Problem 1896 - JEE Physics

Question

Two wires of same length are shaped into a square and a circle. If they carry same current, ratio of the magnetic moment is (a) \(2: \pi\) (b) \(\pi: 2\) (c) \(\pi: 4\) (d) \(4: \pi\)

Step-by-Step Solution

Answer

The ratio of the magnetic moment of the square to the circle is \(\pi: 2\), which corresponds to option (b).

Step 1: Calculate the magnetic moment of the square wire

Let's denote the length of the wire as L. Since the wire is shaped into a square, each side of the square will have a length of L/4. The area A_s of the square can be calculated as: \[A_s = (\frac{L}{4})^2\] Now, let's denote the current passing through the wire as I. The magnetic moment M_s of the square can be calculated as: \[M_s = I \cdot A_s\] By substituting the area of the square, we get: \[M_s = I \cdot (\frac{L}{4})^2\]

Step 2: Calculate the magnetic moment of the circle wire

To find the magnetic moment of the circle wire, we first need to determine its radius. Since the circumference of the circle is equal to the length of the wire, we have: \[2\pi r = L\] Solving for the radius r, we get: \[r = \frac{L}{2\pi}\] Now, we can calculate the area A_c of the circle as: \[A_c = \pi r^2\] Substituting the radius, we get: \[A_c = \pi (\frac{L}{2\pi})^2\] The magnetic moment M_c of the circle can be calculated as: \[M_c = I \cdot A_c\] By substituting the area of the circle, we get: \[M_c = I \cdot \pi (\frac{L}{2\pi})^2\]

Step 3: Calculate the ratio of the magnetic moments

We need to find the ratio of the magnetic moments of the square and the circle: \[\frac{M_s}{M_c} = \frac{I \cdot (\frac{L}{4})^2}{I \cdot \pi (\frac{L}{2\pi})^2}\] We can see that the current I and the length L will cancel out, so the ratio of the magnetic moments simplifies to: \[\frac{M_s}{M_c} = \frac{(\frac{1}{4})^2}{\pi (\frac{1}{2\pi})^2}\] After calculating the ratio, we arrive at the following: \[\frac{M_s}{M_c} = \frac{1}{16} \cdot \frac{4\pi^2}{\pi}\] This further simplifies to: \[\frac{M_s}{M_c} = \frac{\pi}{2}\] So the ratio of the magnetic moment of the square to the circle is \(\pi: 2\), which corresponds to option (b).

Key Concepts

Square Wire

A square wire is formed when a wire is bent to create a four-sided figure with equal sides. When dealing with the magnetic moment, understanding the geometry of the square is crucial. The magnetic moment of a current-carrying wire depends on the area enclosed by the wire.
For a square, each side has the same length. If the total length of the wire is denoted as \( L \), each side of the square will therefore be \( \frac{L}{4} \).

• Area of the square, \( A_s = (\frac{L}{4})^2 \)


• Magnetic moment, \( M_s = I \cdot A_s \)

This means you multiply the area of the square by the current flowing through the wire to find its magnetic moment. The shape of the wire directly influences the calculation of its magnetic properties.

Circle Wire

A circle wire is obtained when a wire is shaped into a circle, forming a closed loop. This shape is vital in magnetic moment calculations as the circular form provides a different area compared to the square.
The circumference of a circle wire equals the total length of the wire, \( L \), giving us \( 2\pi r = L \) where \( r \) is the radius.

• Solve to find the radius: \( r = \frac{L}{2\pi} \)


• Area of the circle, \( A_c = \pi r^2 \)


• Substituting \( r \), \( A_c = \pi (\frac{L}{2\pi})^2 \)

Finally, find the magnetic moment of the circle wire by multiplying the current \( I \) with its area: \( M_c = I \cdot A_c \). This calculation depicts how the circular geometry affects the magnetic properties of the wire.

Current

Current, represented as \( I \), is the flow of electric charge in a wire. It is a fundamental component when determining the magnetic moment of any wire.
The magnetic moment \( M \) is directly proportional to the current and the area encapsulated by the wire.

• Square wire: \( M_s = I \cdot (\frac{L}{4})^2 \)


• Circle wire: \( M_c = I \cdot \pi (\frac{L}{2\pi})^2 \)

Both expressions highlight the role of current in these calculations. Regardless of wire shape, the current remains a constant factor influencing the wire’s magnetic characteristics. Understanding the path and direction of current is key in predicting both circular and square wire properties.

Area of Shapes

The area of shapes plays a pivotal role in calculating the magnetic moment for wires of different configurations.
For a square wire, the area is simple to compute as the square comprises readily defined straight lines:

• Calculate area, \( A_s = (\frac{L}{4})^2 \)

In contrast, the circle wire has a curved shape, meaning the area is dependent on the radius derived from the wire's total length:

• Calculate area, \( A_c = \pi (\frac{L}{2\pi})^2 \)

The difference in calculating these areas underscores the importance of understanding geometric properties when dealing with shapes. This knowledge is essential in the persistent influence they exert on physical phenomena like the magnetic moment. Different shapes lead to different magnetic characteristics even if the original wire length and current remain the same.