Question

To practice Tactics Box 25.1 Using Lenz's law. Lenz's law is a useful rule for determining the direction of the induced current in a loop. Specifically, it says that there is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux. The following Tactics Box summarizes the essential steps in using Lenz's law.

TACTICS BOX 25.1 Using Lenz's law

1. Determine the direction of the applied magnetic field. The field must pass through the loop.

2. Determine how the flux is changing. Is it increasing, decreasing, or staying the same?

3. Determine the direction of an induced magnetic field that will oppose the change in the flux:

• Increasing flux: The induced magnetic field points opposite the applied magnetic field.

• Decreasing flux: The induced magnetic field points in the same direction as the applied magnetic field.

• Steady flux: There is no induced magnetic field.

4. Determine the direction of the induced current. Use the right-hand rule to determine the current direction in the loop that generates the induced magnetic field you found in step 3.

Follow the steps above to solve the problem: A wire carries a current in the direction indicated in the figure. A loop is located next to the wire. If the current in the wire increases, is there a clockwise current around the loop, a counterclockwise current, or no current? (Figure 1)

Part A) Determine the direction of the magnetic field of the current-carrying wire.   

Part B) As the current in the wire increases, the magnetic field around the wire increases in magnitude without changing its direction. Does this affect the flux through the loop, and if so, how?

Answer 1

Concepts and reason

The concepts used in this problem are right hand rule of electromagnetism, magnetic flux passing through an area and Lenz’s law of electromagnetic induction.

First, determine the direction of the magnetic field of the current carrying wire using right hand rule of magnetic field.

Later, obtain the expression for magnetic flux relating magnetic field and area. Then, using this expression, find out whether the magnetic flux through the loop increases or decreases.

Finally, determine the direction of induced current in the loop by using Lenz’s law of electromagnetic induction.

Fundamentals

The right-hand rule of electromagnetism can be explained as below,

Point your right hand’s thumb along the flow of the current of positive charge. Curl your fingers as they are wrapped around the conductor. In this case, your fingers point the direction of magnetic field.

The magnetic flux passing through an area is expressed as below,

$\begin{array}{l}\\{\Phi _B} = B \cdot A\\\\{\Phi _B} = BA\cos \theta \\\end{array}$

Here, $B$ is the magnitude of the magnetic field, $A$ is area through which the magnetic field is passing and $\theta$ is the angle between magnetic field and unit vector of the area.

When unit vector of the area and magnetic field are directed in the same direction, then angle between them is zero.

Substitute, $0$ for $\theta$ in the above equation ${\Phi _B} = BA\cos \theta$ .

$\begin{array}{l}\\{\Phi _B} = BA\cos 0\\\\{\Phi _B} = BA\left( 1 \right)\\\\{\Phi _B} = BA\\\end{array}$

Lenz’s law of electromagnetic induction states that, magnetic field produced by induced magnetic field opposes the applied magnetic field.

Thus when applied magnetic field increases, induced magnetic field points opposite the applied magnetic field.

Thus when applied magnetic field decreases, induced magnetic field points in the same direction as the applied magnetic field.

(A)

Apply right hand rule of electromagnetism to the current carrying wire.

Point your right hand’s thumb along the flow of the current of positive charge. Curl you fingers as they are wrapped around the conductor.

In this case, the thumb denotes the direction of the current, which is upward. The wrapped fingers denote the direction of magnetic field.

The above figure shows that the current( $I$ ) flowing the wire is directed upward and the induced magnetic field( $B$ ) is directed into the page.

The direction of the magnetic field passing through the loop is into the page.

(B)

The expression for the magnetic field is as below,

${\Phi _B} = BA$

The above equation clearly shows that, the magnetic flux is directly proportional to the applied magnetic field and area of the loop through which the magnetic field passes.

As the current ( $I$ ) increases, magnitude of the magnetic field ( $B$ ) increases, which result in increases of the magnetic flux ( ${\Phi _B}$ ) by above equation.

(C)

Apply Lenz’s law of electromagnetic induction to the loop.

Applied magnetic field B_app is into the page, The induced magnetic field B_ind will oppose B_app and hence B_ind will have opposite direction of B_app. Thus direction of B_ind is out of the page.

Now apply right hand rule of electromagnetism.

Now curl your fingers of right hand around the wire in such a way that figers point the direction of induced magnetic field. In this case the thumb shows the direction of the current flowing through the loop, as shown in the figure below.

The above figure shows that, applied magnetic field is directed into the page while the induced magnetic field is directed out of the page. The current induced by the induced magnetic field is directed counterclockwise around the loop.

Ans: Part A

The magnetic field is directed into the page.

Part B

As the magnetic field increases the magnetic flux increases.

Part C

The current flows counterclockwise around the loop.