Section 9: Magnetic Forces on Moving Charges In this lesson you will derive an expression for the magnetic force caused by a current carrying conductor on another current carrying conductor apply F = BIL sinθ from Lesson 7 to define one ampere derive F = qvb sinθ for the magnetic force on a single moving charge apply the concepts to numerical exercises use a left hand rule to determine the direction of the magnetic force on the moving charge 1
Review Biot s Law : Strength of Magnetic Field around a Straight Conductor Motor Principle: Force Experienced by a Current Carrying Conductor in a Magnetic Field F = BIL sinθ, explains the basic principle of the electric motor. In the first picture there is no mention of a magnetic force. But in the second picture the magnetic force is a very important outcome. A word of caution Do not think that a Biot's Law situation will never lead to a force situation. If you have two Biot's Law set up side by side, then the field of one wire could play the role of the permanent field. 2
Using Magnetic Fields to define one Ampere Recall: One ampere of current represents a flow of 1 coulomb of charge in one second and that 1 coulomb of charge is comprised of more than 6 billion, billion electrons! Obviously nobody counted this many electrons, so this definition of an ampere is not very elegant. We are going to improve on this situation by using the work of Ampere and Biot. To do so we mount two current carrying wires in the same vicinity. Furthermore, we will assume that each wire carries 1.0 A of current. The direction of the current is not important, but for the sake of the discussion, let's say one flows upward (towards the ceiling) and the other flows downward (towards the floor). While the direction of the current is not critical, we are going to be very specific about the length of the wires and the distance that the wires are apart. Specifically, the wires are 1.0 m long and are also 1.0 m apart. Finally, since our set up is in air, the magnetic permeability is μ o = 4π x 10 7 T m/a. 3
All of the foregoing is summarized in the picture below. After all, I 1 = I 2 = 1.0 A, therefore B 1 = B 2. The magnitudes of F 2 on 1 and F 1 on 2 would be equal even if the currents were different (due to Newton's Third Law). The important thing to notice is that in the area between the wires, the fields are in the same direction (out of the screen) and therefore repel each other. Now the job is to compute the magnitude of the repelling force (F). You have to decide which magnetic field (B 1 or B 2 ) will play the role of the "permanent" field. It doesn't make any difference, so let's say B 1 is the permanent field. Remember Section 7: According to the motor principle, wire #2 will experience a force because it is carrying a current and lying in the magnetic field B 1. That force is F = F 1 on 2 = B 1 I 2 L 2 sin θ, where sin θ = 1 because the downward current in wire #2 is perpendicular to the horizontal "circles" of B 1. This leaves us with F = B 1 I 2 L 2. It's important to notice that the F is the force that wire #1 is exerting on wire #2. The right hand side of the expression tells us this as well. There you will see that the magnetic field of wire #1 (that's B 1 ) is acting on the current and length of wire #2 (that's I 2 and L 2 ). The expression F = B 1 I 2 L 2 is not of much use because even though we know I 2 and L 2, we do not have the equipment to measure B 1. This is where Biot comes in. Remember from Section 8 that the following expression can be written for wire #1: where r is the distance from wire #1 to wire #2 and μ o is the magnetic permeability of free space (in this case air). Substituting Biot's Law into F = B 1 I 2 L 2 gives where the right hand side includes only quantities that we know. 4
At last we can calculate the magnetic force that two 1.0 m wires exert on each other when they each carry a current of of 1.0 A and are 1.0 m apart. Here goes: Units Recall [T] = [N/A m] And finally we can write a more satisfying definition of one ampere of current: One ampere (1 A) is the current flowing through two parallel wires placed one meter apart in air when the wires exert a force of 2 x 10 7 N/m on each other for each metre of their length. 5
Example Two wires are lying parallel to each other and 13 mm apart. One of the wires is carrying a current of 3.5 A and the force between the wires is 2.5 x 10 4 N/m. What is the current in the other wire? The Magnetic Force on a Single Moving Charge We have been saying that the force on a current carrying conductor as it sits an external magnetic field is F = BIL sinθ. The key phrase here is current carrying. If there were no current, there would be no force on the conductor (unless the conductor was made of iron or some other magnetic material). So, the force is really exerted on the current. The conductor just provides a path for the current. If a stream of electrons were shot through a magnetic field the force on the stream would still be F = BIL sinθ, where, in this case, L would be the length of the stream that falls within the magnetic field. 6
If we are interested in the force on a single charge we can represent that charge by q. If we consider a stream of such particles moving a length of L in a time of t we could represent its speed by v = L/t or, better still, L = vt. Recall as well that current, I, is defined as charge/time, so, I = nq/t. Substituting these variables in our force equation But this is the force on n charges. On a single charge, n = 1. The force for a single charge therefore becomes F = Bqv sin θ Summing up, the magnetic force on an individual moving charge is given by the equation F = Bqv sin θ where B is the magnetic field strength in tesla (T), q is the magnitude of the charge in coulombs (C) that is moving at a velocity v in m/s, and θ is the angle between v and B. To determine the direction of the force on a negative charge that is passing through a magnetic field, just apply left hand rule #3: point your fingers in the direction of the magnetic field, your thumb in the direction that the charge is moving, then the palm of your hand points in the direction of the force on the charge. 7
Examples 1. A magnetic field of 44.0 T is directed into the screen. A particle with a negative charge of 2.0 x 10 18 C is shot into the field from the right, making an angle of 90 o with the field lines. If the particle is moving at 5.4 x 10 7 m/s, what magnetic force does it experience? 2. An electron travels with a speed of 2.00 x 10 6 m/s in a plane perpendicular to a 1.00 x 10 3 T magnetic field. What is the radius of the electron s path? (m e = 9.11 x 10 31 kg.) Page 652, Questions 3a, 5a, 6a, 7, 8 8
9
10