Chapter 21, Problem 15P

To determine

Part a:

The external force required to move the rod.

Answer to Problem 15P

Solution:

Magnetic force on current in moving bar, $\frac{B^2{l}^{2}v}{R}$.

Explanation of Solution

Given:

Formula used:

The magnitude of the induced EMF is as follows:

$\epsilon =Blv$

Here, B is the magnetic field, l is the length, and v is the velocity.

The expression for the induced Current is as follows:

$I=\frac{\epsilon }{R}$

Here, $R$ is the resistance.

The expression for the force is as follows:

$F=IBl$

Here, I is the current.

Calculation:

According to the Lenz’s law the induced current produced in the loop due to the change in the magnetic flux. The changing magnetic flux produces an induced magnetic field, which produces an induced EMF.

As the rod is pulled away from the uniform magnetic field, an induced EMF is produced.

From the Lenz’s law the induced EMF is directed into the page due to the decreasing inward magnetic flux through the loop and this results a clockwise induced current produced in the loop.

Due to an induced current in the left side of loop, there will be a magnetic force acting on it directed to the left. This opposing force must be equal to the force directed to the right to keep the rod moving with constant velocity.

Substitute $\frac{\epsilon }{R}$ for I and $Blv$ for $\epsilon$ in the equation $F=IBl$.

$\begin{array}{c}F=\left(\frac{\epsilon }{R}\right)Bl\\ =\frac{B^2{l}^{2}v}{R}\end{array}$

Therefore, the force is as follows:

$F=\frac{B^2{l}^{2}v}{R}$

To determine

Part b:

The external power required to move the rod.

Answer to Problem 15P

Solution:

The external power required to move the rod is $\frac{B^2{l}^2{v}^2}{R}$.

Explanation of Solution

Given:

Formula used:

The expression for the external power required is as follows:

$P_{\text{Ex}\text{.}}=Fv$

Here, F is the external force needed to move the rod and v is the speed with which the rod moves.

Calculation:

Substitute $\frac{B^2{l}^{2}v}{R}$ for F in the equation $P_{\text{Ex}\text{.}}=Fv$.

$\begin{array}{c}{P}_{\text{Ex}\text{.}}=\left(\frac{B^2{l}^{2}v}{R}\right)v\\ =\frac{B^2{l}^2{v}^2}{R}\end{array}$