Question

The magnetic field at the centre of a circular current carrying loop of radius 12 cm is 6.4 × 10^-6 T. What is the magnetic moment of the loop?

Answer 1

Data:

R = 12 cm = 12 × 10^-2 m

B = 6.4 × 10^-6 T

µ₀ = 4π x 10^-7 T·m/A

Formula:

The magnetic moment (M) of a current-carrying loop is given by the formula:

M = NIA

The magnetic field at the center of a circular current-carrying loop is given by the formula:

B = `(mu_0I)/(2R)`

Rearrange the formula to solve for `I = (2Br)/(mu_0)`

Solution:

⇒ `I = (2 xx (6.4 xx 10^-6) xx 0.12)/(4pi xx 10^-7)`

⇒ `I = (1.536 xx 10^-6)/(4pi xx 10^-7)`

⇒ `I = (1.536 xx 10^-6)/(4 xx 3.14 xx 10^-7)`

⇒ `I = (1.536 xx 10^-6)/(4 xx 3.14 xx 10^-7)       ...(pi = 3.14)`

⇒ `I = (1.536 xx 10^-6)/(12.56 xx 10^-7)`

⇒ `I = (1.536 xx 10^-6)/(1.256 xx 10^-6)`

⇒ `I = (1.536)/(1.256)`

⇒ `I = 1.223 A`

∴ The area of the circular loop: 

A = `pir^2`      ...(Formula)

⇒ A = 3.14 × (0.12)²

⇒ A = 3.14 × 0.12 × 0.12

⇒ A = 3.14 × 0.0144

⇒ A = 0.0452 m²

∴ M = NIA        ...(Formula)

⇒ M = 1 × 1.223 x 0.0452

⇒ M = 0.0552 A.m²

∴ The magnetic moment of the loop is 0.0552 A·m².