Question

Two circular coils of radius 'a' and '2a' are placed coaxially at a distance 'x' and '2x' respectively from the origin along the X-axis. If their planes are parallel to each other and perpendicular to the X-axis and both carry the same current in the same direction, then the ratio of the magnetic field induction at the origin due to the smaller coil to that of the bigger one is:

• A. 2 : 1
• B. 1 : 1
• C. 1 : 4
• D. 1 : 2

Hint:

When solving for the magnetic field produced by a coil at a point along its axis, use the formula derived for a circular loop of current. Pay attention to the geometry and distances involved, and simplify accordingly to find the ratio of fields.

Answer 1

The Correct Option is A

For a circular coil of radius \(r\) carrying current \(I\), the magnetic field at a point on the axis of the coil at a distance \(x\) from the center of the coil is given by the formula: \[ B = \frac{{\mu_0 I r^2}}{{2 (r^2 + x^2)^{3/2}}} \] Where: 
- \( B \) is the magnetic field at the point on the axis, 
- \( \mu_0 \) is the permeability of free space, - \( I \) is the current in the coil, 
- \( r \) is the radius of the coil, - \( x \) is the distance from the center of the coil to the point on the axis. 
Now, applying this formula to both coils: 
1. For the smaller coil with radius \(a\) and distance \(x\) from the origin, the magnetic field at the origin is: \[ B_{\text{small}} = \frac{{\mu_0 I a^2}}{{2 (a^2 + x^2)^{3/2}}} \] 
2. For the larger coil with radius \(2a\) and distance \(2x\) from the origin, the magnetic field at the origin is: \[ B_{\text{large}} = \frac{{\mu_0 I (2a)^2}}{{2 ((2a)^2 + (2x)^2)^{3/2}}} \] Now, simplifying the ratio \( \frac{B_{\text{small}}}{B_{\text{large}}} \): \[ \frac{B_{\text{small}}}{B_{\text{large}}} = \frac{{a^2 (a^2 + x^2)^{3/2}}}{{(2a)^2 ((2a)^2 + (2x)^2)^{3/2}}} \] Upon simplifying, we get: \[ \frac{B_{\text{small}}}{B_{\text{large}}} = 2 \] Thus, the ratio of the magnetic field at the origin due to the smaller coil to that of the bigger one is \( 2 : 1 \).