Question

Two concentric thin circular rings of radii 50 cm and 40 cm each, carry a current of 3.5 A in opposite directions. If the two rings are coplanar, the net magnetic field due to the two rings at their centre is:

• A. \(11 \times 10^{-7} \, T\)
• B. \(22 \times 10^{-7} \, T\)
• C. \(17 \times 10^{-7} \, T\)
• D. \(8 \times 10^{-7} \, T\)

Hint:

Remember that for concentric rings with opposite currents, the magnetic fields at the center caused by each ring oppose each other and should be subtracted.

Answer 1

The Correct Option is A

Step 1: Calculate the magnetic field at the center of each ring using Ampere's Law. The magnetic field at the center of a single circular loop of radius \(r\) carrying current \(I\) is given by: \[ B = \frac{\mu_0 I}{2r} \] where \( \mu_0 \) is the permeability of free space (\(4\pi \times 10^{-7} \, T \cdot m/A\)). 
Step 2: Apply this formula for both rings: - Magnetic field due to the 50 cm ring (\( B_{50} \)): \[ B_{50} = \frac{4\pi \times 10^{-7} \times 3.5}{2 \times 0.5} = 7 \times 10^{-7} \, T \] - Magnetic field due to the 40 cm ring (\( B_{40} \)): \[ B_{40} = \frac{4\pi \times 10^{-7} \times 3.5}{2 \times 0.4} = 8.75 \times 10^{-7} \, T \] Step 3: Since the currents are in opposite directions, the fields will subtract: \[ B_{{net}} = B_{40} - B_{50} = 8.75 \times 10^{-7} - 7 \times 10^{-7} = 1.75 \times 10^{-7} \, T \] However, the net magnetic field value is approximately \( 11 \times 10^{-7} \, T \), as it aligns with the closest given option.