Magnetic Effects & Magnetism

JEE Main level, single correct, medium

66 of 276

Question 66 of 276

The fractional change in the magnetic field intensity at a distance rr from centre on the axis of current carrying coil of radius aa to the magnetic field intensity at the centre of the same coil is: (Take r<ar\lt a)

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Answer: (C) 32r2a2\dfrac{3}{2}\dfrac{{r}^{2}}{{a}^{2}}

As we know,
Baxis=μ0iR22(R2+x2)3/2where,R=a{B}_{axis}=\dfrac{{\mu }_{0}{iR}^{2}}{2{\left({R}^{2}+{x}^{2}\right)}^{3/2}}\text{where},R=a
Bcentre=μ0i2a{B}_{\text{centre}}=\dfrac{{\mu }_{0}i}{2a}
∴Baxis=μ0ia22(a2+r2)3/2∴{B}_{\text{axis}}=\dfrac{{\mu }_{0}{ia}^{2}}{2{\left({a}^{2}+{r}^{2}\right)}^{3/2}}
∴∴ fractional change in magnetic field =μ0i2a−μ0ia22(a2+r2)3/2μ0i2a=1−1[1+(r2a2)]3/2=\dfrac{\dfrac{{\mu }_{0}i}{2a}-\dfrac{{\mu }_{0}i{a}^{2}}{2{\left({a}^{2}+{r}^{2}\right)}^{3/2}}}{\dfrac{{\mu }_{0}i}{2a}}=1-\dfrac{1}{{\left[1+\left(\dfrac{{r}^{2}}{{a}^{2}}\right)\right]}^{3/2}}
≈1−[1−32r2a2]=32r2a2\approx 1-\left[1-\dfrac{3}{2}\dfrac{{r}^{2}}{{a}^{2}}\right]=\dfrac{3}{2}\dfrac{{r}^{2}}{{a}^{2}}

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