Question

Two concentric and coplanar circular loops P and Q have their radii in the ratio 2:3. Loop Q carries a current 9 A in the anticlockwise direction. For the magnetic field to be zero at the common centre, loop P must carry ______.

Options

• 3 A in clockwise direction

• 9 A in clockwise direction

• 6 A in anti-clockwise direction

• 6 A in the clockwise direction

Answer 1

Two concentric and coplanar circular loops P and Q have their radii in the ratio 2:3. Loop Q carries a current 9 A in the anticlockwise direction. For the magnetic field to be zero at the common centre, loop P must carry 6 A in the clockwise direction.

Explanation:

Given: The ratio of the radius of the loops P and Q = 2:3

Current in the loop Q = 9 A in (anticlockwise direction)

To Find: The current in loop P for which the magnetic field at the common centre becomes zero.

⇒  The intensity of the magnetic field (B) at the centre of a circular current-carrying coil is given by the formula:

Magnetic Field (B) = `(μ_0i)/(2R)`

• Where 'i' is the current flowing through the circular coil.
• Here 'R' is the radius of the circular coil.
• 'μ₀' is a constant known as the permeability constant of free space.

⇒ The direction of the magnetic field due to the circular current-carrying coil is given by the right-hand thumb rule.

According to this law, if we curl our right-hand palm around the current-carrying loop with fingers pointing in the direction of the current flow, then the direction of our right-hand thumb will give us the direction of the Magnetic field.

⇒ In the given question, the current in loop Q is flowing in the anticlockwise direction. Therefore for the resultant magnetic field to be zero at the common centre, the current in the loop P must flow in the clockwise direction.

Let the magnetic field due to the loop P be 'B₁'

Let the magnetic field due to the loop Q be 'B₂'

Let the current through the loop P be 'i₁'

The ratio of the radius of the loops P and Q is R₁:R₂ is equal to 2:3.

⇒ The magnetic field 'B₁' due to the current-carrying loop P is given by:

B₁ = `(μ_0i_1)/(2R_1)` – Equation (i)

⇒ The magnetic field 'B₂' due to the current-carrying loop Q is given by:

B₂ = `(μ_0i_2)/(2R_2)` – Equation (ii)

⇒ Equating the equations (i) and (ii):

∵ B₁ = B₂

∴ `(μ_0i_1)/(2R_1) = (μ_0i_2)/(2R_2)`

∴ `i_1/i_2 = R_1/R_2`

∴ `i_1 = 2/3 xx i_2`

∴ `i_1 = 2/3 xx 9`

∴ i₁ = 6 Ampere

Therefore for the magnetic field to be zero at the common centre, the loop P must carry​ a current of 6 Ampere in the clockwise direction.