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Aggressive behaviour may occur and is usually undirected or resistive and the mania spasms compilation discount 100 mg pletal with visa. Several case series have reported a preponderance of patients with complex partial epilepsy spasms back muscles order discount pletal, patient is likely to spasms under rib cage purchase pletal once a day be amnesic for the event back spasms 20 weeks pregnant order 100mg pletal free shipping. Post-ictal depression can last longer Inter-ictal psychosis (up to two weeks) than other post-ictal states. Symptoms range from mild to severe and may involve the prevalence is reported to be 4-10% in patients with epilepsy, mainly in those with temporal lobe suicidal behaviour. It has been reported to occur more commonly with right-sided temporal or frontal epilepsy7,8,9. It is a chronic disorder and clinically resembles chronic schizophrenia (symptoms of foci2. There are a few case reports of post-ictal mania delusions, hallucinations, thought disorder) but there are some reports that personality is better preserved. The risk factors that have been reported are early age of onset of epilepsy, bilateral temporal foci and a refractory course. The Post-ictal psychosis pathophysiological mechanisms of psychosis in epilepsy are unclear and both focal and generalised brain the prevalence has been estimated to be 6-10% in patients with epilepsy, particularly temporal lobe abnormalities have been implicated12–15. It typically occurs after a cluster of complex partial seizures (+/ secondary generalisation). There is usually a period of lucidity (12-72 hours) prior to the onset of psychosis. The psychotic symptoms Treatment with antipsychotic medications is usually long term. The atypical antipsychotic drugs are include delusions, hallucinations, thought disorder or mania, which are usually transient but can last potentially less likely to reduce seizure threshold (with the exception of clozapine) or cause extrapyramidal several weeks. It has also been reported that some patients with recurrent episodes of post-ictal psychosis side effects. Predisposing risk factors are ictal fear, bilateral epileptic foci or support and family education are also important. Mechanisms are unknown but may be related to transient neurochemical changes as a result of seizures. Treatment-related psychiatric problems Treatment of acute post-ictal psychosis may require short courses of benzodiazepines or antipsychotics. Phenobarbitone, primidone, tiagabine, Inter-ictal topiramate, vigabatrin and felbamate have been associated with depression. Depression Research has shown that nearly 40% of patients studied in tertiary epilepsy centres had major depression Improved seizure control has been associated with the emergence of psychiatric symptoms. The true prevalence of depression introduced the term ‘forced normalisation’ which refers to a dramatic reduction in epileptiform activity in epileptic patients in the community has not been established. The risk may be higher in patients who are on polytherapy, become seizure However, it is important to recognise that some patients can present with atypical depressive symptoms, free abruptly, or if there is a past psychiatric history. This is characterised by chronic intermittent dysthymia, irritability and anxiety symptoms. Epilepsy surgery Transient mood disturbances (emotional lability, depression and anxiety) have been reported following Treatment for depression includes psychological interventions such as counselling, psychotherapy or temporal lobe surgery for epilepsy (about 25%) in the first 6-12 weeks16. For moderate to severe depression, antidepressant medications (10%), symptoms, particularly depression, may persist and require psychiatric treatment. Electroconvulsive treatment can be effective for severe medication-resistant depression but post-surgical psychiatric evaluation to form part of the assessment/management for epilepsy surgery. Inter-ictal anxiety disorders References the incidence of inter-ictal anxiety disorders is greater than in the general population. Postictal psychiatric events during prolonged video are reportedly more common in patients with temporal lobe epilepsy, especially with left-sided foci. Dysphoric disorders and paroxysmal affects: recognition and treatment of epilepsy-related psychiatric disorders. Schizophrenia-like psychosis and epilepsy: the status of the association [review]. Psychosis associated with epilepsy: significance of the laterality of the epileptogenic lesion. A ‘mock up’ of schizophrenia: temporal lobe epilepsy and schizophrenia-like psychosis. Epilepsy, psychosis, and schizophrenia: clinical and neuropathologic correlations. A magnetization transfer imaging study in patients with temporal lobe epilepsy and interictal psychosis. A prospective study of the early postsurgical psychiatric associations of epilepsy surgery. Nevertheless, the diagnosis of epilepsy is frequently straightforward, particularly when precise and detailed personal and eyewitness accounts of the prodrome, onset, evolution and recovery period after the event are obtained. Misdiagnosis is common, however, and possibly affects up to 2-30% of adults with a diagnosis of epilepsy1,2. This and other reports highlight the high rate of misdiagnosis of epilepsy, the cause of which is undoubtedly multifactorial. The reasons for misdiagnosis may include a deficiency of relevant semiological information obtained during the ascertainment of the clinical history, lack of understanding of the significance of specific clinical features and over-reliance on the diagnostic value of routine investigations4. The attainment of a correct diagnosis is of paramount importance as an erroneous diagnosis of epilepsy has physical, psychosocial5 and socioeconomic consequences for the patient, and economic implications for the health and welfare services6. Syncope Transient loss of awareness is common, and may affect up to 50% of people at some stage of life7,8,9. Elucidating the aetiological basis for an episode of loss of awareness is challenging. Typically, the episode is transient, patients are generally unable to provide an accurate description of the event and there may be a lack of reliable witnesses, particularly in the elderly who, more frequently, live alone. The difficulty in establishing an accurate diagnosis is further hampered by systemic and neurological examinations and subsequent investigations frequently being normal after an episode or between habitual attacks when the patient is seen in the hospital ward or clinic10. Orthostatic syncope is caused by autonomic failure rather than an exaggerated and inappropriate but Syncope is more prevalent than either epilepsy or dissociative (psychogenic) seizures and is common essentially normal physiological response, as seen in neurocardiogenic syncope. Vasovagal syncope is vasoconstrictor response to standing, resulting in venous pooling and a postural fall in blood pressure, most frequently encountered in adolescence, whereas syncope due to cardiac causes becomes increasingly usually within seconds or minutes of becoming upright. The annual incidence of syncope in the elderly population in long-term stays warm and well perfused, the pulse rate is unchanged and sweating is absent. Recurrence is not unusual, occurring in approximately dysfunction are varied and include autonomic neuropathy due to diabetes, alcohol, amyloidosis, genetic 30% of patients, typically within the first two years after symptom onset13. Recurrence is associated abnormalities or complex autonomic failure, such as primary autonomic failure or multiple system with increased morbidity, such as fractures, subdural haematomas and soft-tissue injuries14, and impaired atrophy. Medications such as antihypertensives, phenothiazines, tricyclic antidepressants, diuretics and quality of life11. There are numerous causes of syncope, each resulting in inappropriate systemic hypotension and critical Postural orthostatic tachycardia syndrome cerebral hypoperfusion. It arises through the provocation Even mild stimulation to the neck results in presyncopal symptoms or syncope from marked bradycardia of inappropriate reflex hypotension, with a variable degree of bradycardia, or even transient asystole. There may be a family history of ‘fainting’ or recent addition 30% of elderly patients with unexplained syncope and drop attacks22,23. It is of vasoactive medication targeted at, for example, hypertension or ischaemic heart disease. A typical attack commences with prodromal symptoms of nausea, clammy sweating, blurring or greying Around 30% of cases are classified as cardioinhibitory where the predominant manifestations are sinus visual impairment, lightheadedness, and ringing or roaring tinnitus. Occasionally, visual and auditory bradycardia, atrioventricular block, or asystole due to vagal action on sinus and atrioventricular nodes. Many of these individual symptoms Permanent pacemaker implantation is effective at reducing recurrence rate24. The vasodepressor type also are difficult for patients to describe and their description may be vague, but collectively the cluster comprises 30% of cases and results in a marked decrease in vasomotor tone without a change in heart rate. Untreated symptomatic patients have a syncope recurrence bradycardia and acral paraesthesia may be present. Muscle tone is reduced, causing the eyes to roll up, rate as high as 62% within four years.

The most significant symptom is weakness occurring with exertion that rapidly improves upon rest back spasms 6 weeks pregnant purchase pletal us. Drugs to spasms when falling asleep purchase 50 mg pletal amex avoid in myasthenia: — Aminoglycosides — Magnesium — Succinylcholine Final Diagnosis Myasthenia gravis 767 Case 10 Chief Complaint “My wife was weak in her right hand and couldn’t speak or write spasms gallbladder pletal 100mg on line. He explains that muscle spasms 72885 generic 100mg pletal overnight delivery, although his wife is usually quite verbal, she had difficulty expressing her thoughts. The husband was unable to understand what she was saying, so he gave her a pen and pad, but she could only write a couple of words which didn’t make much sense. Although hypertension has the strongest correlation with the risk of stroke, other risks include diabetes, smoking, and hyperlipidemia. Even without a cardiac murmur, it is wise to obtain an echocardiogram, particularly if no other source can be found for an embolus. Aspirin is used most often, but in cases of aspirin intolerance, clopidogrel may be used. Other antiplatelent combinations are clopidogrel + aspirin or aspirin + dipyridamole. A worsening deficit is presumed to be a clot in formation, and this is why heparin is used. The patient initially noted the onset of numbness and weakness in the legs while walking downhill. He has started to sit at bus stops waiting for the pain to resolve and then walks to the next bus stop where he stops again for a rest. The numbness and tingling increase with walking, and are alleviated by sitting and lying on his side. You instruct the patient to lie on the examination table with his hips and knees flexed. Spondylolisthesis (displaced vertebra) Initial Management Setting: outpatient Diagnostic/Therapeutic Plan Ankle-brachial index Peripheral pulses Test Results Ankle-brachial index: normal (1. Ankle-brachial index is an excellent first test to determine if symptoms are caused by compromised blood supply. In addition, claudication is usually bilateral, and the pain and abnormalities on physical exam should have no relation to body position. Also, pain due to “pseudo” claudication by lumbar spinal stenosis is often increased by walking downhill and relieved by walking uphill. Patients with lumbar spinal stenosis often describe relief from pain when walking flexed with a shopping cart. That is different from the pain associated with a herniated disc, which usually worsens with flexion and is linked to a history of chronic back pain. A displaced vertebra in spondylolisthesis may or may not be palplated on physical exam, but this differential can be ruled out with imaging. In claudication, the exercise tolerance should be predictable and reproducible, whereas with spondylosis, the pain can be variable. Over the past week, she noted an ascending numbness and tingling in her legs, followed by difficulties walking. Physical examination shows a pale optic disc with color desaturation in the right eye. Reflexes are increased in the lower extremities with a positive Babinski response bilaterally. Individually, these deficits could represent strokes, encephalitis, vasculitis, etc. The most common presentation is fatigue with focal sensory symptoms and gait disturbances. As in this patient, optic neuritis can be the first demyelinating event in approximately 20% of cases, which most often resolves spontaneously. It is caused by damage of the medial longitudinal fasciculus, a heavily-myelinated tract which allows conjugate eye movement by connecting the paramedian pontine reticular formation with the abducens nucleus complex of the contralateral side to the oculomotor nucleus of the ipsilateral side. The patient is not able to adduct the affected eye when trying to look at the contralateral side with the unaffected eye showing a corrective nystagmus. Also, in acute disseminated encephalomyelitis, the multiple lesions develop simultaneously and there generally should be no recurrence. More commonly, however, the main presentation is associated with the rash of erythema migrans and bilateral facial nerve palsy. Because the best explanation for the disease is an autoimmune phenomenon, treatment is largely based on immune-modulating drugs, such as steroids, adrenocorticotrophic hormones, or disease-modifying agents. Disease-modifying agents are used for long-term therapy to prevent relapse and slow the progression of the disease. Final Diagnosis Multiple sclerosis 786 Case 13 Chief Complaint “My husband walks with a shuffle. He and his wife initially attributed his problem to the natural course of aging, but now he seems to be just generally slow. There is a resting tremor, which resolves with movement, and cogwheel rigidity with paratonia. Important differential diagnoses include essential tremor, progressive supranuclear palsy, secondary parkinsonism and other neurodegenerative disorders. Progressive supranuclear palsy is characterized by patients presenting with rigidity and dystonic postures of the neck and shoulders, and a tendency to topple while walking. Paralysis of vertical gaze and eventually, lateral gaze distinguishes it from Parkinson’s disease. Tolcapone and entacapone are useful as L dopa extenders (both are ineffective when 790 used alone) For equivocal cases, serial clinical examinations are warranted Discussion the standard treatment for Parkinson’s disease is the replacement of the neurotransmitter dopamine with the precursor L-dopa (L-hydroxyphenylalanine. Anticholinergic agents such as trihexyphenidyl and benztropine mesylate are also used in secondary parkinsonism resulting from medications. Selegiline increases concentrations of dopamine by blocking metabolism via inhibition of the enzyme monoamine oxidase. A thorough history will bring out important features of this syndrome long before the patient or spouse suspects that something is wrong. Bradykinesia or slowness in both the initiation and execution of movement is also characteristic of this disease. Final Diagnosis Parkinson’s disease Basic Science Correlate Biochemistry Tyrosine hydroxylase is the rate-limiting enzyme for dopamine production. Pharmacology Loss of dopamine in the striatal pathway is the main factor driving Parkinson’s symptoms, so treatment strategies focus on raising dopamine availability through different mechanisms. She describes visual symptoms which begin in the right field of vision and consist of bright, flashing lights. Nausea and vomiting can occur with any acute headache, but it is mostly characteristic of migraines. Methylsergide is reserved for refractory cases because of the risk of cardiac, retroperitoneal, or pulmonary fibrosis. Discussion the birth control pills should be discontinued for this patient, as should the smoking, as they can precipitate or worsen migraine attacks. Abortive treatment is used in the acute setting and includes sumatriptan, dihydroergotamine, and ergotamine tartrate, which work as serotonergic agonists to relieve the headaches. Triptans can be given orally, intranasally, or even subcutaneously, depending on the severity of the headache. The classification of headache has some prognostic significance and major importance in determining treatment: Migraines can be bilateral or localizing. In a cluster headache, the pain peaks in 5 minutes, whereas with migraine it takes several hours to peak. The best abortive therapy for an acute attack is a triptan or ergotamine; if those are contraindicated, 100% oxygen inhalation can be effective. Tension headache, described as “bandlike” around the head (like a belt being tightened around the head) is usually bilateral. This patient’s intermittent symptoms—with a normal exam between events—argue against a space-occupying lesion, especially given the history of intermittent symptoms over years. When the police bring her to the emergency department, the patient is angry and resistant. Hyperthyroidism Initial Diagnostic Plan/Test Results Urine toxicology screen: negative Thyroid function tests: normal Assessment the differential diagnosis between bipolar I disorder and a psychotic disorder is sometimes difficult.

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The Nuclei: Black/Blue (Hematoxylin) stain is used for examining exfoliative cells of vaginal, cervical, Catalog Number Volume sputum, as well as other body secretions. The lipoid capsule of the acid-fast organism takes up carbol fuchsin and Control: Intestine or Colon. The Fontana-Masson Stain Kit is intended for use in the histological visualization of Argentaffin cells and Melanin in Acid Mucopolysaccharides: Bright Blue paraffin or frozen sections. Demonstration of Muscle Fibers: Pale Pink elastic tissue is useful in cases of emphysema (atrophy of elastic Erythrocytes: Gray, Yellow or Pink tissue), arteriosclerosis (thinning and loss of elastic fibers) and Rickettsia: Reddish-Purple various other vascular diseases. The Eosinophil Mast Cell Stain Kit is intended for simultaneous Catalog Number Volume visualization of eosinophils and mast cells. Gram Posi ve Bacteria: Blue Advantages include lower evaporation rate, better color Gram Nega ve Bacteria: Red patterns, reduced tendency to spill over container, hands, and Other Tissue: Slightly Yellow Pink countertops, and improved surface tension to remain on tissue Nuclei: Red section. Our Hematoxylin produces crisp, intense blue nuclei Catalog Number Volume providing optimal contrast to the Eosin stained cytoplasm. Helicobacter Pylori has been shown to be the causative Iron Stain Kit organism in some gastric ulcers. Ferric Cytoplasm: Pink to Red iron is normally found in small amounts in bone marrow and the Background: Violet spleen. Abnormally large deposits may be seen in Nuclei: Blue hemochromatosis and hemosiderosis. This product is based on the Prussian Blue reaction in which ionic iron reacts with acid Catalog Number Volume ferrocyanide producing a blue color. Pylori Rapid Stain Kit is designed for demonstrating Background: Pink Helicobactor Pylori infected tissue. Kit may be used on formalin Catalog Number Volume fixed, paraffin-embedded tissue. Jones Stain Kit (For Basement Membrane) Results: Helicobactor Pylori: Blue the Jones Stain Kit is intended for use in histological Mucin: Yellow demonstration of the basement membrane and reticular fibers. Background: Light Blue this procedure is ideal for staining renal glomerular basement membranes. The main function of the basement membrane and Catalog Number Volume reticular fibers is to provide anchorage and support. This product is used for identifying the basic neuronal structure in brain or spinal cord sections. This procedure is particularly valuable for staining a variety of other bodily secretions and cell useful when studying the heart, blood vessels and various smears. Elas c Fibers: Black to Blue/Black Nuclei: Blue Nuclei: Blue/Black High Keratin Cells: Orange Collagen: Yellow Superficial Cells: Pink Re cular Fibers: Yellow Erythrocytes: Dark Pink Mucin: Bright Blue Parabasal Cells: Blue/Green Fibrin: Bright Red Intermediate Cells: Blue/Green Muscle: Red Metaplastic Cells: May contain both Blue/Green and Pink. This product is useful in distinguishing mucin use in histological demonstration of lymphocytes and negative undifferentiated squamous cell lesions from mucin mucopolysaccharides. In addition, this product will stain the glycogen to break it into smaller sugars that are then washed off mucopolysaccharide capsule of Cryptococcus neoformans. The staining patterns of Fat Cells: Red the lymphocytes are helpful in making therapeutic decisions in Neutral Fat: Red established cases of lymphocytic leukemia. The staining pattern of the lymphocytes visualization of Pneumocystis carinii in cytology smears and are helpful in making therapeutic decisions in established cases paraffin tissue sections. Using this procedure easily allows viewing of collagenous this kit is designed to demonstrate many collagen, fibrin, muscle septa as thin as 0. Additionally, this procedure is useful in the assessment of the Erythrocytes: Pink to yellowish-red. This Trichrome Special Stain kit may be used on Catalog Number Volume formalin-fixed, paraffin-embedded or frozen sections. Ammoniacal silver stains are the most commonly used methods for demonstration of reticular fibers. Wright-Giemsa Stain Kit is intended to be used for differential Catalog Number Volume staining of blood smears, bone marrow and blood parasites. When stained, reticulocytes appear to contain dark blue granules Catalog Number Volume or filaments. The number of reticulocytes in the blood is a simple Astra Blue Solution measure of erythropoietic performance. Reticulocytes are Astra Blue Solution is designed to stain mast cells bright blue. Biebrich Scarlet/Acid Fuchsin Solution Erythrocytes: Blue/Green Used in Masson’s Trichrome. Nissl granules are purple, nuclei of neuroglia and capsule of the acid-fast organism takes up carbol fuchsin and endothelial cells are slighly bluer than Nissl granules. Nissl granules are purple, nuclei of neuroglia and endothelial cells are slightly bluer than Nissl granules. The lipoid capsule of the acid-fast organism takes up carbol fuchsin and Stains Nissel granules and basophilia. This newly formulated Eosin Gentian Violet Solution provides the benefits of a traditional alcoholic formula with significant improvements in usability. Gentian Violet is intended for the demonstration and tension to remain on tissue section. Helicobacter Pylori has been shown to be the causative organism in some gastric ulcers. Hematoxylin Solution (5%) Jenner Stock Solution is intended to be used for differential Catalog Number Volume staining of blood smears, bone marrow and blood parasites. Eosinophils: Violet nuclei with dark violet granules and bright red Catalog Number Volume cytoplasmic granules. Product is supplied as a two component (equal correctly, various shades of green can be obtained to aid in volumes of parts A and B) system that is mixed prior to use and visualization of tissue components. In addition, this product will stain the mucopolysaccharide capsule of May-Grunwald Stock Solution Catalog Number Volume Component of the Giemsa Stain Kit (May-Grunwald) for blood smears. This Solution is the component responsible for staining the reagent may be used on formalin-fixed, paraffin-embedded or Erythrocytes yellow. In addition, most formulations develop a small amount of precipitate over Catalog Number Volume extended periods of time. The procedure that is intended for use in histological demonstration stain is used for examining exfoliative cells of vaginal, cervical, of lymphocytes and mucopolysaccharides. Commonly used in of the lymphocytes are helpful in making therapeutic decisions in combination with hematoxylin. Potassium Ferrocyanide Solution (3%) Rhodanine Solution (Stock) Used in the Colloidal Iron staining procedure. Sudan Black B Solution (in Propylene Glycol) Catalog Number Volume Sudan Black B (in Propylene Glycol) is intended for use in the histological visualization of fat cells and neutral fat. This reagent is used in Tartrazine provides a light yellow background to aid in the many special stain procedures including the Colloidal Iron Stain visualization of other stains. A humidity chamber designed by a histotechnologist that eliminates the majority of individual slide handling. 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The scalar product means that we take the component of the force along the direction of the trajectory at each point spasms treatment 50 mg pletal visa. We can make this clearer by writing Z t 2 dx W = F · dt t1 dt the integrand infantile spasms 7 month old cheap 100mg pletal otc, which is the rate of doing work muscle relaxer jokes proven pletal 100mg, is called the power muscle relaxant before massage generic pletal 50 mg, P = F · x. Using Newton’s second law, we can replace F = mx to get Z t Z t 2 1 2 d W = m x · x dt = m (x · x) dt = T (t2) T (t1) t1 2 t1 dt where 1 T m x · x 2 is the kinetic energy. Except in all advanced courses of theoretical physics, kinetic energy is always denoted T which is why I’ve adopted the same notation here). But a simple result (which you will prove in your Vector Calculus course) says that (2. The resulting force also depends only on the distance to the origin and, moreover, always points in the direction of the origin, dV F(r) = rV = x (2. In these lectures, we’ll also use the notation r = x to denote the unit vector pointing radially from the origin to the position of the particle. In the vector calculus course, you will spend some time computing quantities such as rV in spherical polar coordinates. Then, using the chain rule, we have V V V rV =,, x1 x2 x3 dV r dV r dV r =,, dr x1 dr x2 dr x3 dV x1 x2 x3 dV =,, = x dr r r r dr 2. For now, we will just mention what is important about central forces: they have an extra conserved quantity. Let’s look at what happens to angular momentum in the presence of a general force F. We’re left with dL = mx x = x F dt the quantity = x F is called the torque. This gives us an equation for the change of angular momentum that is very similar to Newton’s second law for the change of momentum, dL = dt Now we can see why central forces are special. When the force F lies in the same direction as the position x of the particle, we have x F = 0. This means that the torque vanishes and angular momentum is conserved dL = 0 dt We’ll make good use of this result in Section 4 where we’ll see a number of important examples of central forces. They are • Gravity • Electromagnetism • Strong Nuclear Force • Weak Nuclear Force the two nuclear forces operate only on small scales, comparable, as the name suggests, 15 to the size of the nucleus (r0 10 m). We can’t really give an honest description of these forces without invoking quantum mechanics and, for this reason, we won’t discuss them in this course. It determines the strength of the gravitational force and is given by 11 3 1 2 G 6. We will devote much of Section 4 to studying the motion of a particle under the inverse-square force. The gravitational eld due to many particles is simply the sum of the eld due to each individual particle. If we x particles with masses Mi at positions ri, then the total gravitational eld is X M i (r) = G |r ri| i the gravitational force that a moving particle of mass m experiences in this eld is X M i F = Gm (r ri) |r r |3 i i the Gravitational Field of a Planet the fact that contributions to the Newtonian gravitational potential add in a simple linear fashion has an important consequence: the external gravitational eld of a spher ically symmetric object of mass M – such as a star or planet – is the same as that of a point mass M positioned at the origin. The proof of this statement is an example of the volume r integral that you will learn in the Vector Calculus course. Summing over the contribution from R all points x inside the planet, the gravitational eld is given by Z 3 G(x) (r) = d x Figure 5: |x|R |r x| It’s best to work in spherical polar coordinates and to choose the polar direction, = 0, to lie in the direction of r. We can use this to write an expression 2 2 2 for the denominator: |rx| = r +x 2rx cos. The gravitational eld then becomes Z R Z Z 2 2 (x)x sin (r) = G dx d d v r2 + x2 2rx cos 0 0 0 Z R Z 2 (x)x sin = 2G dx d v r2 + x2 2rx cos 0 0 Z R h i 1 v = 2 2 2 = 2G dx (x)x r + x 2rx cos 0 rx =0 Z R 2G = dx (x)x (|r + x| |r x|) r 0 – 23 – So far this calculation has been done for any point r, whether inside or outside the planet. R2 this is the familiar potential energy that gives rise to constant acceleration. If you want to escape the gravitational attraction of the planet for – 24 – ever, you will need energy E 0. The reason that this is dodgy is because, as we will see in Section 7, the laws of Newtonian physics need modifying for particles close to the speed of light where the eects of special relativity are important. Suppose that the escape velocity from the surface of a star is greater than or equal to the speed of light. Although the derivation above is not trustworthy, by some fortunate coincidence 2 it turns out that the answer is correct. If a star is so dense that it lies within its own Schwarzchild radius, then it will form a black hole. You’ll be pleased to hear that, because both objects are much larger than their Schwarzchild radii, neither is in danger of forming a black hole any time soon. The mass appearing in the second law represents the reluctance of a particle to accelerate under any force. In contrast, the – 25 – mass appearing in the inverse-square law tells us the strength of a particular force, namely gravity. Since these are very dierent concepts, we should really distinguish between the two dierent masses. We now know that the inertial and gravitational masses are equal to within about one part 13 in 10. Currently, the best experiments to study this equivalence, as well as searches for deviations from Newton’s laws at short distances, are being undertaken by a group at the University of Washington in Seattle who go by the name Eot-Wash. Their role – at least for the purposes of this course – is to guide any particle that carries electric charge. The force experienced by a particle with electric charge q is called the Lorentz force, F = q E(x) + x B(x) (2. By convention, particles with positive charge q are accelerated in the direction of the electric eld; those with negative electric charge are accelerated in the opposite direction. Due to a quirk of history, the electron is taken to have a negative charge given by 19 qelectron 1. It is a velocity dependent force, with magnitude proportional to the speed of the particle, but with direction perpendicular to that of the particle. In this case, the electric eld is always of the form E = r For some function (x) called the electric potential (or scalar potential or even just the potential as if we didn’t already have enough things with that name). Claim: the conserved energy is 1 E = mx · x + q(x) 2 Proof: E = mx · x + qr · x = x · (F + qr) = qx · (x B) = 0 where the last equality occurs because x B is necessarily perpendicular to x. Notice that this gives an example of something we promised earlier: a velocity dependent force which conserves energy. The key part of the derivation is that the velocity dependent force is perpendicular to the trajectory of the particle. A particle of charge Q sitting at the origin will set up an electric eld given by Q Q r E = r = (2. The quantity 0 has the grand name Permittivity of Free Space and is a constant given by 12 3 1 2 2 0 8. It is a remarkable fact that, mathematically, the force looks identical to the Newtonian gravitational force (2. We will study motion in this potential in detail in Section 4, with particular focus on the Coulomb force in 4. Although the forces of Newton and Coulomb look the same, there is one important dierence. In contrast, the electro static Coulomb force can be attractive or repulsive because charges q come with both signs. Further dierences between gravity and electromagnetism come when you ask what happens when sources (mass or charge) move; but that’s a story that will be told in dierent courses. The equation of motion is mx = q x B Let’s pick the magnetic eld to lie in the z-direction and write B = (0, 0, B) We can now write the Lorentz force law (2. There are a number of ways to solve them, but a particularly elegant way is to construct the complex variable = x + iy. Circles of ar x bitrary sizes are allowed; the only price that you pay is that you have to go faster. Figure 7: A Comment on Solving Vector Dierential Equa tions the Lorentz force equation (2. The straightforward way to view these is always in components: they are three, coupled, second order dierential equations for x, y and z.

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