# 30.1. The magnetic force

**. The magnetic force between two charges q**

__magnetic force___{1}and q

_{2}, moving with velocities v

_{1}and v

_{2}, is equal to

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_{0}is called the

**which is equal to 4[pi] x 10**

__permeability constant__^{-7}Ns

^{2}/C

^{2}, and r is the distance between the two charges (see Figure 30.1). The ratio R of the magnetic force and the electric force is equal to

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**Figure 30.1. Relevant vectors for definition of magnetic force.**

_{0}and u

_{0}into eq.(30.2), the ratio R can be rewritten as

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^{8}m/s). Clearly, the magnetic force is small compared with the electric force unless the speed of the particles is high (a significant fraction of the velocity of light).

A magnetic field B can be associated with the magnetic force. The magnetic field at some point in the vicinity of a moving charge can be determined by placing a test charge at that point and moving it with some velocity v. The test charge will experience, besides the electric force, a magnetic force F

_{mag}. Per definition, the magnetic field B is related to the magnetic force F

_{mag}via

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**. Comparing eq.(30.1) and eq.(30.4) we can determine the magnetic field generated by a point charge q**

__Tesla (T)___{2}moving with a velocity v

_{2}:

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### Example: Problem 30.10

At the surface of a pulsar, or neutron star, the magnetic field may be as strong as 10^{8}T. Consider the electron in a hydrogen atom on the surface of the neutron star. The electron is at a distance of 0.53 x 10

^{-10}m from the proton and has a speed of 2.2 x 10

^{6}m/s. Compare the electric force that the proton exerts on the electron with the magnetic force that the magnetic field of the neutron star exerts on the electron. Is it reasonable to expect that the hydrogen atom will be strongly deformed by the magnetic field ?

The electron in a hydrogen atom is at a distance r equal to 0.53 x 10

^{-10}m from the proton. The electric force acting on the electron is equal to

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## 30.2. The Biot-Savart Law

**The definition of the magnetic force showed that two moving charges experience a magnetic force. In other words, a moving charge produces a magnetic field which results in a magnetic force acting on all charges moving in this field.**

A current flowing through a wire is equivalent to a collection of electrons moving with a certain velocity along the direction of the wire. Each of the moving electrons produces a magnetic field that is given by eq.(30.5). Consider a small segment of the wire with a length dL (see Figure 30.2). At any given time, a charge dq will be located in this segment. The magnetic field, dB, generated by this charge at point P is equal to

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**Figure 30.2. Calculation of magnetic field produced by an electric current.**

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**.**

__Biot-Savart Law__### Example: Problem 30.33

Helmholtz coils are often used to make reasonably uniform magnetic fields in laboratories. These coils consist of two thin circular rings of wire parallel to each other and on a common axis, the z-axis. The rings have a radius R and they are separated by a distance which is also R. These rings carry equal currents in the same direction. Find the magnetic field at any point on the z-axis.**Figure 30.3. Calculation of magnetic field produced by one ring.**

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To find the field generated by a pair of Helmholtz coils, we assume that the coils are centered at z = 0 and at z = R. The magnetic field generated by the coil located at z = 0 is given by eq.(30.19). The magnetic field generated by the coil located at z = R is given by

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**Figure 30.4. Magnetic field generated by a coil with R = 1 m.**

**Figure 30.5. Magnetic field generated by a pair of Helmholtz coils.**

### Example: Problem 30.22

A very long wire is bent at a right angle near its midpoint. One branch of it lies along the positive x-axis and the other along the positive y-axis (see Figure 30.6). The wire carries a current I. What is the magnetic field at a point in the first quadrant of the x-y plane ?**Figure 30.6. Problem 30.22**

**Figure 30.7. Field generated by wire.**

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## 30.3. The magnetic dipole

**The magnetic field on the axis of a current loop was discussed in Problem 30.33. At large distances from the current loop (z >> R) the field is approximately equal to**

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^{3}. This dependence of the magnetic field strength on distance is similar to the dependence observed for the electric field strength of an electric dipole:

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### Example: Problem 30.44

An amount of charge Q is uniformly distributed over a disk of paper of radius R. The disk spins about its axis with angular velocity [omega]. Find the magnetic dipole moment of the disk.The first step to solve this problem is to determine the dipole moment of a ring of the disk, with radius r and with a width dr. The amount of charge dq on this ring is equal to

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