Question

The magnetic field strength at a point at a distance $d$ from the centre, on the axial line of a very short bar magnet of magnetic moment $M$ is $B$. The magnetic induction at a distance $2d$ from centre, on equatorial line of magnet of magnetic moment $8M$, will be $B/x$. Find $x$.

Answer 1

2

Answer 2

The problem involves calculating the magnetic field strength due to a short bar magnet at different points and relating them.

1. Magnetic Field on the Axial Line: The magnetic field strength ($B_{axial}$) at a point at a distance $r$ from the center on the axial line of a short bar magnet with magnetic moment $M$ is given by: $B_{axial} = \frac{\mu_0}{4\pi} \frac{2M}{r^3}$ According to the problem, at a distance $d$ from the center on the axial line, the magnetic field strength is $B$. So, $B = \frac{\mu_0}{4\pi} \frac{2M}{d^3} \quad \text{(Equation 1)}$

2. Magnetic Field on the Equatorial Line: The magnetic induction ($B_{equatorial}$) at a point at a distance $r$ from the center on the equatorial line of a short bar magnet with magnetic moment $M$ is given by: $B_{equatorial} = \frac{\mu_0}{4\pi} \frac{M}{r^3}$ According to the problem, we need to find the magnetic induction at a distance $2d$ from the center on the equatorial line of a magnet with magnetic moment $8M$. Let this new magnetic induction be $B'$. Substitute $r = 2d$ and $M = 8M$ into the equatorial field formula: $B' = \frac{\mu_0}{4\pi} \frac{8M}{(2d)^3}$ $B' = \frac{\mu_0}{4\pi} \frac{8M}{8d^3}$ $B' = \frac{\mu_0}{4\pi} \frac{M}{d^3} \quad \text{(Equation 2)}$

3. Relate B and B' to find x: The problem states that the new magnetic induction $B'$ is equal to $B/x$. $B' = \frac{B}{x}$ Now, substitute the expressions for $B$ from Equation 1 and $B'$ from Equation 2: $\frac{\mu_0}{4\pi} \frac{M}{d^3} = \frac{1}{x} \left( \frac{\mu_0}{4\pi} \frac{2M}{d^3} \right)$ To solve for $x$, we can cancel out the common terms $\left( \frac{\mu_0}{4\pi} \frac{M}{d^3} \right)$ from both sides of the equation: $1 = \frac{1}{x} \times 2$ $1 = \frac{2}{x}$ $x = 2$

The value of $x$ is 2.