The glowing Northern Lights occur when charged particles move in the Earth's magnetic field. Let's look at the magnetic forces that act on the charges. First consider three charged particles: one stationary with respect to the magnet, the second moving parallel to the magnetic field lines, and the third moving across the magnetic field lines. Which of these charged particles do you think will interact with the magnetic field and be given a push by the field? ... No force is exerted on charged particles that are stationary relative to the magnetic field. ... As long as a charged particle moves parallel to field lines, the magnetic field exerts no force on it. ... When a charged particle moves across magnetic field lines, the force of the field acts on the particle. ... It is interesting that a force is exerted on the charged particle by the magnetic field only when a particle moves across the field lines. In other words, when a charged particle is at rest or moves parallel to the magnetic field, the force exerted by the magnetic field on the charged particle is zero. How do you think that the speed of the particle affects the force exerted on it? How will the magnitude and direction of the force differ for a charged particle moving quickly across the fields lines and one moving more slowly? ... Consider two more factors: the magnitude of the charge and the strength of the magnetic field. Which combination of charge and field strength do you think causes the greatest force? ... So far we've found that the force acting on a charged particle is maximized when the charge of the particle and magnetic field are maximized and when the charged particle moves across the magnetic field lines. The direction of the force is perpendicular to both the velocity and the magnetic field vectors. These observations can be summarized by the formula: Force equals charge, q, times velocity, v, times the strength of the magnetic, B. This formula is true only when the motion of the particle is perpendicular to the magnetic field lines. This formula represents the maximum force a magnetic field can exert on a charged particle that is moving through it. What is the force on a six microCoulomb charged particle moving at four times 10 to the fourth meters per second perpendicular to a 2.55 Tesla magnetic field? Check your calculations. ... Note that the units work out nicely to yield Newtons as the unit of force. ... (6.00 x 10-6 C)(4.00 x 104 m/s)(2.55 T) ... But what if the direction of the particle's velocity is not perpendicular to the magnetic field lines? In that case we, must find the component of the velocity that is perpendicular to the magnetic field. This component results in a factor of sine theta in the equation, because v sine theta is the component of the velocity that is at right angles to the magnetic field. The angle theta is easy to find, since it is the angle between the magnetic field and velocity vectors. The formula thus becomes: F = qvB sine theta. Another way of representing this is Force equals charge times the cross product of velocity with the magnetic field strength, or F equals q v cross B. The cross product is also known as the vector product. This name indicates that the result of the calculation is itself a vector, namely, the force vector. When the velocity vector and the magnetic field are not perpendicular, the cross product is the component of velocity that is perpendicular to the field. A two times 10 to the negative fourth gram particle with a charge of two Coulombs enters a 4 Tesla magnetic field at an angle of 30.0 degrees. What is the force on the particle? ... The last step is to determine the direction of the force. ... There is a good way to determine the direction of the force. We know that the velocity, the magnetic field, and the force are all perpendicular to each other, just like the axes are in an x, y, z coordinate system. You can use your right hand to find the direction of the force. Place your hand as shown, and only use your thumb, index finger, and middle finger. The three should be held at right angles to each other. The thumb represents the force. The index finger represents the direction of the velocity vector. And, the middle finger represents the magnetic field. This is known as the right hand rule and its is very helpful in figuring out the direction of the force. Be careful though - if you use your left hand this won't work! Lets go back to our formula F equals the cross product of qv and B. If q is positive, F points in the direction of v cross B. The right hand rule that we just discussed applies. If q is negative, F points in the direction opposite that of v cross B. So a positive charge results in F pointing up, and a negative charge results in F pointing down. Use the right hand rule to determine the direction of the force that will act on this charged particle. Which direction will the force point? ... When a charged particle travels in a direction perpendicular to the magnetic field, the particle moves in a circular path. If the magnetic field is directed into the page, the force always points towards the center of the circle. If the particle is positively charged it moves counter-clockwise; if the particle is negatively charged, it moves clockwise. Using Newton's second law, Force equals mass times acceleration, we can determine the radius of the curved path. Since the particle moves in a circle, its acceleration is equal to its velocity squared divided by its radius. The force is given by the charge q, times velocity v, times the strength of the magnetic field B. By solving, this equation we find an expression for the radius of curvature. A two times 10 to the negative fourth gram particle with a charge of two Coulombs enters a 4 Tesla magnetic field at an angle of 30.0 degrees. What is the radius of the circular path of the particle around the magnetic field lines? ... This can be done by shooting a particle of unknown charge into a known magnetic field and measuring its radius of curvature. We will look at this device that performs this operation, called a mass spectrometer, at the end of this lesson. What path will a charged particle follow if its velocity is not perpendicular to the magnetic field? In other words, if the particle travels at an angle to the magnetic field. The velocity vector can be broken into a perpendicular component, v-perpendicular, and a parallel component, v-parallel. We know that a particle moving along the field line experiences no force. Therefore, the parallel component remains constant. We learned that when a particle travels in a direction perpendicular to the field line, its path is circular. We can put these bits of information together to deduce that the path of a particle traveling at an angle to the magnetic field is helical or spiral. We can make charged particles move with constant velocity through a magnetic field by applying a combination of an electric field and a magnetic field, by using a velocity selector. The total force is called the Lorentz force, and can be written as force equals to charge q, times the strength of the electric field E, plus the cross product of the charge, q, times the velocity, v, and the magnetic field, B. A pair of oppositely charged parallel plates provides the electric field. The electric force, qE, directed downward balances the upward directed magnetic force, qvB, and makes the particles move through the field in a straight line with a constant velocity v = E/B. Suppose we have a 5 tesla magnetic field that is directed into the screen, a 2 volt per meter electric field that is directed from top to bottom, and a particle with 1 coulomb charge traveling through the fields. What speed must the particle have to allow it to travel in a straight line through the fields? ...

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