Question

A wire of length 100 cm is bent in the form of a circular coil of 5 turns and another wire of length 60 cm is bent in the form another circular coil of 4 turns. If the same current flows through the two coils, then the ratio of the magnetic fields induced at the centres of the two coils is:

• A. 20 : 21
• B. 5 : 6
• C. 15 : 16
• D. 3 : 4

Hint:

Calculate the radius for each coil to find the field ratio, considering the same current flows through both.

Answer 1

The Correct Option is C

Let \(L_1\) and \(L_2\) be the lengths of the wires, and \(n_1\) and \(n_2\) be the number of turns for the first and second coils respectively. 
We are given: 
For coil 1: \(L_1 = 100 \, \text{cm}\), \(n_1 = 5\) For coil 2: \(L_2 = 60 \, \text{cm}\), \(n_2 = 4\) Let \(r_1\) and \(r_2\) be the radii of the first and second coils. 
The length of wire used to form a circular coil of \(n\) turns and radius \(r\) is \(L = n \times 2\pi r\). For the first coil: \[ L_1 = n_1 \times 2\pi r_1 \] \[ 100 = 5 \times 2\pi r_1 \] \[ r_1 = \frac{100}{10\pi} = \frac{10}{\pi} \, \text{cm} \] For the second coil: \[ L_2 = n_2 \times 2\pi r_2 \] \[ 60 = 4 \times 2\pi r_2 \] \[ r_2 = \frac{60}{8\pi} = \frac{15}{2\pi} \, \text{cm} \] The magnetic field at the center of a circular coil is given by \(B = \frac{\mu_0 n I}{2r}\). 
Let \(B_1\) and \(B_2\) be the magnetic fields at the centers of the first and second coils respectively, assuming the same current \(I\) flows through both coils. For the first coil: \[ B_1 = \frac{\mu_0 n_1 I}{2r_1} = \frac{\mu_0 \times 5 \times I}{2 \times \frac{10}{\pi}} = \frac{5 \mu_0 I \pi}{20} = \frac{\pi \mu_0 I}{4} \] For the second coil: \[ B_2 = \frac{\mu_0 n_2 I}{2r_2} = \frac{\mu_0 \times 4 \times I}{2 \times \frac{15}{2\pi}} = \frac{4 \mu_0 I \times 2\pi}{30} = \frac{8 \pi \mu_0 I}{30} = \frac{4 \pi \mu_0 I}{15} \] 
The ratio of the magnetic fields is: \[ \frac{B_1}{B_2} = \frac{\frac{\pi \mu_0 I}{4}}{\frac{4 \pi \mu_0 I}{15}} = \frac{\pi \mu_0 I}{4} \times \frac{15}{4 \pi \mu_0 I} = \frac{15}{16} \] Thus, the ratio of the magnetic fields induced at the centers of the two coils is 15:16. 
Correct Answer: (3) 15: 16