Chapter 30, Problem 1Q

If the circular conductor in Fig. 30-21 undergoes thermal expansion while it is in a uniform magnetic field, a current is induced clockwise around it. Is the magnetic field directed into or out of the page?

Figure 30-21 Question 1.

To determine

To find:

The direction of the magnetic field.

Answer to Problem 1Q

Solution:

The direction of the magnetic field is out of the page.

Explanation of Solution

1) Concept:

When the magnetic flux through an area bounded by the conducting loop changes, it induces the current in the loop. The induced current produces its own magnetic field. The direction of this current and the field is decided using Lenz’s law.

2) Formula:

$E=-\frac{d\varnothing }{dt}$

3) Given:

i) The circular conductor undergoes thermal expansion.

ii) The conductor is placed in a uniform magnetic field.

iii) The induced current is in the clockwise direction.

4) Calculations:

The thermal expansion changes the area of the conductor. This changes the flux through the conductor as it is placed in a uniform magnetic field. The change in magnetic flux induces the clockwise current in the conductor. This current produces another magnetic field. The direction of this induced magnetic field is decided by the right hand rule. According to the rule, the direction of the induced magnetic field will be into the page.

According to the Lenz’s law, the direction of this induced magnetic field is such that it opposes the flux change. i.e. the direction of the original field.

Hence, the direction of the original field is out of the page.

Conclusion:

The changing magnetic flux induces current in the conductor. The right hand rule decides the direction of the induced magnetic field. Lenz’s law helps us determine the direction of the original magnetic field since the induced magnetic field opposes the original magnetic field.