Chapter 32, Problem 1PQ

A constant magnetic field of 0.275 T points through a circular loop of wire with radius 3.50 cm as shown in Figure P32.1.

a. What is the magnetic flux through the loop?

b. Is a current induced in the loop? Explain.

FIGURE P32.1

To determine

The magnetic flux through the loop.

Answer to Problem 1PQ

The magnetic flux through the loop is $5.29\times {10}^{-4}\text{Wb}$.

Explanation of Solution

Write the expression for the magnetic flux through the loop.

    ${\varphi }_{\text{B}}=BA\cos \phi$                                                                                                      (I)

Here, ${\varphi }_{\text{B}}$ is the magnetic flux, $B$ is the magnetic field and $\phi$ is the angle between the area vector and the magnetic field.

Write the expression for the area of the loop.

    $A=\pi r^2$

Here, $A$ is the area of the loop and $r$ is the radius of the loop.

Substitute the above equation in equation (I) to find ${\varphi }_{\text{B}}$.

    ${\varphi }_{\text{B}}=B\left(\pi r^2\right)\cos \phi$

Substitute $0.275\text{T}$ for $B$, $3.50\text{cm}$ for $r$ and $60°$ for $\phi$ in the above equation to find ${\varphi }_{\text{B}}$.

    $\begin{array}{c}{\varphi }_{\text{B}}=0.275\text{T}\times 3.14\times {\left(3.50\text{cm}\times \frac{{10}^{-2}\text{m}}{1\text{cm}}\right)}^2\times \cos 60°\\ =5.29\times {10}^{-4}\text{Wb}\end{array}$

Conclusion:

Thus, the magnetic flux through the loop is $5.29\times {10}^{-4}\text{Wb}$.

To determine

Whether the current induced in the loop.

Answer to Problem 1PQ

No current is induced in the loop because there is no change in the flux passing through the loop.

Explanation of Solution

Since the magnetic flux linked with the loop is constant, there is no change in the magnetic flux associated with the loop of the wire. So, there is no current induced in the loop.

Conclusion:

Thus, no current is induced in the loop.