Question

Two identical coils of 5 turns each carry 1 A and 2 A current respectively. Assume they have common centre with their planes parallel to each other. If their radius is 1 m each and the direction of flow of current in the coils are in opposite directions, then the magnetic field produced on its axial line at a distance of \(\sqrt{3}\) m from the common centre is (in tesla)

• A. \(0\)
• B. \(\frac{15}{16}\mu_0\)
• C. \(\frac{8}{16}\mu_0\)
• D. \(\frac{5}{16}\mu_0\)
• E. \(\frac{16}{5}\mu_0\)

Hint:

Opposite currents → subtract magnetic fields.

Answer 1

The Correct Option is D

Concept: Magnetic field on the axis of a circular coil: \[ B = \frac{\mu_0 n I R^2}{2(R^2 + x^2)^{3/2}} \]

Step 1: Write given data.
\[ n = 5, \quad R = 1\,m, \quad x = \sqrt{3}\,m \]

Step 2: Compute denominator term. \[ R^2 + x^2 = 1^2 + (\sqrt{3})^2 = 1 + 3 = 4 \] \[ (R^2 + x^2)^{3/2} = 4^{3/2} = (2)^3 = 8 \]

Step 3: Magnetic field due to each coil. \[ B = \frac{\mu_0 \cdot 5 \cdot I \cdot 1}{2 \cdot 8} = \frac{5\mu_0 I}{16} \]

Step 4: Fields due to both coils. For currents: \[ I_1 = 1A \Rightarrow B_1 = \frac{5\mu_0}{16} \] \[ I_2 = 2A \Rightarrow B_2 = \frac{10\mu_0}{16} \]

Step 5: Direction consideration.
Currents are opposite → magnetic fields oppose. \[ B_{\text{net}} = B_2 - B_1 = \frac{10\mu_0}{16} - \frac{5\mu_0}{16} \]

Step 6: Final result. \[ B = \frac{5\mu_0}{16} \] \[ \boxed{\frac{5}{16}\mu_0} \]