Question

The radius of a circular current loop is made double and the current is made half. The magnetic moment of the loop will become

• A. Halved
• B. Doubled
• C. Four times large
• D. None of these

Hint:

For problems involving proportional changes, you can analyze the dependencies. Here, \(\mu \propto I\) and \(\mu \propto r^2\). So, \(\mu \propto I r^2\). The new moment \(\mu'\) will be proportional to \(I' (r')^2 = (\frac{1}{2}I)(2r)^2 = (\frac{1}{2})(4)Ir^2 = 2Ir^2\). Thus, the new moment is twice the original.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The magnetic dipole moment (\(\mu\)) of a current loop is a vector quantity that measures the strength and orientation of the magnetic field produced by the loop. Its magnitude is the product of the current flowing in the loop and the area enclosed by the loop.
Step 2: Key Formula or Approach:
The formula for the magnetic moment (\(\mu\)) of a current loop is: \[ \mu = I \cdot A \] where \(I\) is the current and \(A\) is the area of the loop.
For a circular loop of radius \(r\), the area is \(A = \pi r^2\).
Therefore, the formula becomes: \[ \mu = I \pi r^2 \] Step 3: Detailed Explanation:
Let the initial conditions be: Initial current = \(I\)
Initial radius = \(r\)
So, the initial magnetic moment is \(\mu_{\text{initial}} = I \pi r^2\).
Now, the conditions are changed as follows: New current, \(I' = \frac{I}{2}\) (halved)
New radius, \(r' = 2r\) (doubled)
Let's calculate the new magnetic moment (\(\mu_{\text{new}}\)) with these new values: \[ \mu_{\text{new}} = I' \cdot \pi (r')^2 \] Substitute the expressions for \(I'\) and \(r'\): \[ \mu_{\text{new}} = \left(\frac{I}{2}\right) \cdot \pi (2r)^2 \] \[ \mu_{\text{new}} = \left(\frac{I}{2}\right) \cdot \pi (4r^2) \] Rearrange the terms: \[ \mu_{\text{new}} = \left(\frac{4}{2}\right) \cdot (I \pi r^2) \] \[ \mu_{\text{new}} = 2 \cdot (I \pi r^2) \] Since \(\mu_{\text{initial}} = I \pi r^2\), we can write: \[ \mu_{\text{new}} = 2 \cdot \mu_{\text{initial}} \] Step 4: Final Answer:
The new magnetic moment is twice the initial magnetic moment. Therefore, the magnetic moment of the loop will be doubled. Option (B) is correct.