Magnetic Effects Of Current And Magnetism

Step 1: Understanding the magnetic field and moment relationship.
The magnetic moment of a current-carrying loop is given by: where is the current and is the area of the loop. The magnetic field at the center of the loop is related to the current and area by: Thus, the current can be written as:
Step 2: Substituting in the magnetic moment formula.
Now, substituting for in the magnetic moment formula, we get:
Step 3: Conclusion.
The magnetic moment is , which is option (B).

Step 1: Magnetic field at the centre of a circular coil.
For a coil of radius with turns carrying current :

Step 2: Case of one loop.
Let the total length of wire be . Then,

Step 3: Case of four loops.
Each loop has length :
Number of turns .
Step 4: Magnetic field for four loops.

Step 5: Taking ratio of fields.

Step 6: Substituting radii.

Step 7: Final result.

Step 1: Stefan–Boltzmann law.
Rate of radiation:

Step 2: Surface area dependence.

Step 3: Comparing old and new conditions.

Step 4: Simplifying.

Step 5: Conclusion.