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A long wire carrying a steady current I ...

A long wire carrying a steady current I lies in the plane of circular conducting loop placed at a certain distance from the wire. There will be an induced current in the loop if it is

A

moved parallel to the wire

B

roatated about an axis perpendicular to the plane of the loop passing through the centre of the loop

C

rotated about an axis parallel to the wire passing through the centre of the loop

D

moved away from the wire

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the conditions under which an induced current will occur in a circular conducting loop due to a long wire carrying a steady current. ### Step-by-Step Solution: 1. **Understanding the Setup**: - We have a long straight wire carrying a steady current \( I \). - A circular conducting loop is placed at a distance \( R \) from the wire. 2. **Magnetic Field Due to the Wire**: - The magnetic field \( B \) at a distance \( R \) from a long straight wire carrying current \( I \) is given by: \[ B = \frac{\mu_0 I}{2 \pi R} \] - Here, \( \mu_0 \) is the permeability of free space. 3. **Induced EMF and Current**: - The induced EMF \( \mathcal{E} \) in the loop can be calculated using Faraday's law of electromagnetic induction: \[ \mathcal{E} = -\frac{d\Phi}{dt} \] - Where \( \Phi \) is the magnetic flux through the loop, given by: \[ \Phi = B \cdot A \] - \( A \) is the area of the loop. 4. **Analyzing Each Option**: - **Option A**: If the loop is moved parallel to the wire, the distance \( R \) remains constant. Therefore, the magnetic field \( B \) and the flux \( \Phi \) do not change. Thus, there is no induced EMF and no induced current. **(Incorrect)** - **Option B**: If the loop is rotated about an axis perpendicular to the plane of the loop, the area remains constant and the angle between the area vector and the magnetic field does not change. Hence, the flux \( \Phi \) is constant, leading to no induced EMF and no induced current. **(Incorrect)** - **Option C**: If the loop is rotated about an axis parallel to the wire, the angle between the area vector of the loop and the magnetic field changes. This results in a change in magnetic flux \( \Phi \), which induces an EMF and consequently an induced current in the loop. **(Correct)** - **Option D**: If the loop is moved away from the wire, the distance \( R \) increases, leading to a decrease in the magnetic field \( B \). This change in magnetic field results in a change in flux \( \Phi \), inducing an EMF and an induced current in the loop. **(Correct)** 5. **Conclusion**: - The induced current in the loop occurs in options C and D. Therefore, the correct answer is that there will be an induced current in the loop if it is either rotated about an axis parallel to the wire (Option C) or moved away from the wire (Option D). ### Final Answer: The induced current in the loop will occur if it is: - **Option C**: Rotated about an axis parallel to the wire. - **Option D**: Moved away from the wire.
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