JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Magnetic Effects of Current

The fractional change in the magnetic field intensity at a distance '

r

' from centre on the axis of current carrying coil of radius '

a

' to the magnetic field intensity at the centre of the same coil is : (Take

r < a

)

Answer: (D)

Solution

On the axis of the coil and at its centre

B_{r} = \frac{μ _{0} I a ^{2}}{2 ( a ^{2} + r ^{2} ) ^{3/2}}, B_{0} = \frac{μ _{0} I}{2 a}

\frac{B _{r}}{B _{0}} = \frac{a ^{3}}{( a ^{2} + r ^{2} ) ^{3/2}} = (1 + \frac{r ^{2}}{a ^{2}})^{- 3/2}

For

r < a

the binomial expansion gives

(1 + \frac{r ^{2}}{a ^{2}})^{- 3/2} \approx 1 - \frac{3 r ^{2}}{2 a ^{2}}

, and the fractional change is

\frac{B _{0} - B _{r}}{B _{0}} = 1 - \frac{B _{r}}{B _{0}} = \frac{3}{2} \frac{r ^{2}}{a ^{2}}

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option