Question

The ratio of magnetic field and magnetic moment at the centre of a current carrying circular loop is $x$. When both the current and radius is doubled the ratio will be

• A. \(\frac{x }{ 8}\)
• B. \(\frac{x }{ 4}\)
• C. \(\frac{x }{ 2}\)
• D. $2x$

Answer 1

The Correct Option is A

The magnetic field at the centre of a current carrying loop is given by
\(B=\frac{\mu_{0}}{4 \pi}\left(\frac{2 \pi i}{a}\right)=\frac{\mu_{0} i}{2 a}\)
The magnetic moment at the centre of current carrying loop is given by \(M=i\left(\pi a^{2}\right)\)
Thus, \(\frac{B}{M}=\frac{\mu_{0} \dot{i}}{2 a} \times \frac{1}{i \pi a^{2}}=\frac{\mu_{0}}{2 \pi a^{3}}=x\) (given)
When both the current and the radius are doubled, the ratio becomes
\(\frac{\mu_{0}}{2 \pi(2 a)^{3}}=\frac{\mu_{0}}{8\left(2 \pi a^{3}\right)}=\frac{x}{8}\)

So, the correct option is (A) : \(\frac{x}{8}\).

Answer 2

The magnetic field \( B \) at the center of a current-carrying circular coil is given by the formula: \[ B = \frac{\mu_0 I}{2r} \] Where: - \( B \) is the magnetic field at the center, - \( \mu_0 \) is the permeability of free space, - \( I \) is the current, - \( r \) is the radius of the coil. The magnetic moment \( M \) of the coil is given by: \[ M = I \pi r^2 \] The ratio of the magnetic field \( B \) to the magnetic moment \( M \) is: \[ \frac{B}{M} = \frac{\frac{\mu_0 I}{2r}}{I \pi r^2} = \frac{\mu_0}{2 \pi r^3} \] Now, if both the current \( I \) and the radius \( r \) are doubled, the new values for current and radius are \( 2I \) and \( 2r \) respectively. Substituting these new values into the formulas: The new magnetic field will be: \[ B_{\text{new}} = \frac{\mu_0 (2I)}{2(2r)} = \frac{\mu_0 I}{2r} \] The new magnetic moment will be: \[ M_{\text{new}} = (2I) \pi (2r)^2 = 8 I \pi r^2 \] Now, the new ratio of magnetic field to magnetic moment is: \[ \frac{B_{\text{new}}}{M_{\text{new}}} = \frac{\frac{\mu_0 I}{2r}}{8 I \pi r^2} = \frac{\mu_0}{16 \pi r^3} \] Comparing this with the original ratio: \[ \frac{B}{M} = \frac{\mu_0}{2 \pi r^3} \] We see that the new ratio is \( \frac{1}{8} \) of the original ratio, so the new ratio becomes \( \frac{x}{8} \).

Thus, the correct answer is \( \frac{x}{8} \), corresponding to option (D).