Question

A circular coil of radius R carries an electric current. The magnetic field due to the coil at a point on the axis of the coil located at a distance r from the center of the coil, such that $r >> R$, varies as
$A. \dfrac {1}{r}$
$B. \dfrac { 1 }{ { r }^{ { 3 }/{ 2 } } }$
$C. \dfrac { 1 }{ { r }^{ 2 } }$
$D. \dfrac { 1 }{ { r }^{ 3 } }$

Answer 1

To solve this problem, use Biot-Savart law. Biot-Savart law helps to determine magnetic fields produced by an electric current. Use the formula for Biot-Savart law for the magnetic field at a point on the axis of the loop. Substitute the given condition which is given as $r >> R$, in the above-mentioned formula. Evaluate the expression and find the relationship between magnetic field B and distance r. This will give the variation of magnetic field at a point on the axis of the coil with distance r.

Formula used: $B=\dfrac { { \mu }_{ 0 } }{ 4\pi } \dfrac { 2\pi I{ R }^{ 2 } }{ { \left( { r }^{ 2 }+{ R }^{ 2 } \right) }^{ { 3 }/{ 2 } } }$

Complete step by step answer:
Magnetic field at a point on the axis of a loop is given by,
$B=\dfrac { { \mu }_{ 0 } }{ 4\pi } \dfrac { 2\pi I{ R }^{ 2 } }{ { \left( { r }^{ 2 }+{ R }^{ 2 } \right) }^{ { 3 }/{ 2 } } }$ …(1)
Area of a circular loop is given by,
$A= \pi {R}^{2}$ …(2)
Substituting equation. (2) in equation. (1) we get,
$B=\dfrac { { \mu }_{ 0 } }{ 4\pi } \dfrac { 2IA }{ { \left( { r }^{ 2 }+{ R }^{ 2 } \right) }^{ { 3 }/{ 2 } } }$
$\Rightarrow B=\dfrac { { \mu }_{ 0 } }{ 2\pi } \dfrac { IA }{ { \left( { r }^{ 2 }+{ R }^{ 2 } \right) }^{ { 3 }/{ 2 } } }$ …(3)
When $r>>R$, equation. (3) becomes,
$B=\dfrac { { \mu }_{ 0 } }{ 2\pi } \dfrac { IA }{ { \left( { r }^{ 2 } \right) }^{ { 3 }/{ 2 } } }$
$\Rightarrow B=\dfrac { { \mu }_{ 0 } }{ 2\pi } \dfrac { IA }{ { r }^{ 3 } }$ …(4)
From the equation (4), we can infer that the magnetic field is proportional to the cube of the distance r.
$\Rightarrow B\propto \dfrac { 1 }{ { r }^{ 3 } }$
Thus, the magnetic field due to the coil at a point on the axis of the coil located at a distance r from the center of the coil, such that $r >> R$, varies as $\dfrac { 1 }{ { r }^{ 3 } }$.

So, the correct answer is “Option D”.

Note: Students should remember the Biot-Savart law, it helps to solve these types of problems. If we want to find the magnetic field at the center of the current loop then equation. (1) can be used. Substituting r=0 will give the expression for the magnetic field at the center of the current loop. At the center of the coil, the magnetic field will be uniform, As the distance of the point increases from the center of the coil, the magnetic field decreases.