Answer
There are three magnetic force equations given in chapter 20.
Work Step by Step
a. The first equation is
$$F=I\mathcal{l}Bsin\theta$$
This is the force on a segment of current-carrying wire in a magnetic field.
The current is measured in amperes, the length in meters, and the magnetic field in tesla. A tesla is a newton per ampere - meter.
Here’s how the units work.
$$(A)(m)(\frac{N}{A\cdot m})=(N)$$
b. The second equation is
$$F=qvBsin\theta$$
This is the force on a moving, charged particle in a magnetic field.
The charge is measured in coulombs, the speed in meters per second, and the magnetic field in teslas.
A tesla is a newton per ampere - meter. A coulomb per second is an ampere.
Here’s how the units work.
$$(C)(\frac{m}{s})(\frac{N}{A\cdot m})=(\frac{C}{s})(m)(\frac{N}{A\cdot m})= (N)$$
c. The third equation is
$$F=\frac{\mu_o I_1 I_2}{2\pi d}\mathcal{l}_2$$
This is the force on a length $\mathcal{l}_2$ of wire carrying current $I_2$ due to a parallel long straight wire carrying current $I_1$. The permeability of free space has units of tesla-meter per amp.
Here’s how the units work.
$$ (\frac{T\cdot m}{A})(A)(A)(m)\frac{1}{ m}$$
$$=(T\cdot m)(A) =(\frac{N}{A\cdot m}\cdot m)(A)=(N)$$
