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Chapter 15: Problem 7

A circular coil \(A\) has a radius \(R\) and the current flowing through it is \(I\). Another circular coil \(B\) has radius \(2 R\) and if \(2 I\) is the current flowing through it, then the magnetic field at the centre of the circular coil are in the ratio of (A) \(4: 1\) (B) \(2: 1\) (C) \(3: 1\) (D) \(1: 1\)

Short Answer

Expert verified
The ratio of the magnetic fields at the center of the circular coils A and B is 2:1. The correct answer is (B) 2:1.

Step by step solution

01

Calculate magnetic field at the center of coil A

To find the magnetic field at the center of coil A, we'll use the formula: \[B_A = \frac{\mu_0 I}{2R}\] Here, 碌鈧� is the permeability of free space, which is a constant and has the value of \( 4 \pi \times 10^{-7} \frac{Wb}{A m} \). Since we only want the ratio of the magnetic fields, it's not necessary to plug in the value of 碌鈧�. We'll keep it as 碌鈧� for our calculations.
02

Calculate magnetic field at the center of coil B

For coil B, we're given the radius is twice that of coil A, and the current is also double. So the required magnetic field at the center of coil B will be: \[B_B = \frac{\mu_0 (2I)}{2(2R)}\]
03

Find the ratio of the magnetic fields

Now we have the magnetic fields at the center of both the coils: \[ B_A = \frac{\mu_0 I}{2R} \text{ and } B_B = \frac{\mu_0 (2I)}{2(2R)} \] To find the ratio of the magnetic fields, we'll divide B_A by B_B: \[\frac{B_A}{B_B} = \frac{\frac{\mu_0 I}{2R}}{\frac{\mu_0 (2I)}{2(2R)}}\] We can simplify this expression as: \[\frac{B_A}{B_B} = \frac{ \cancel{\mu_0} \cancel{I} \cancel{2} (2\cancel{R}) }{ \cancel{2} \cancel{R}(2\cancel{I}\cancel{\mu_0})} = \frac{2}{1}\]
04

Identify the correct answer option

The ratio of the magnetic fields at the center of the circular coils A and B is 2:1. Looking back at the options given in the exercise, we find that the correct answer is (B) 2:1.

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Most popular questions from this chapter

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