A coaxial cable consists of an inner wire of radius 'a' surrounded by an outer shell of inner and outer radii 'b' and 'c' respectively. The inner wire carries an electric current i0, which is distributed uniformly across cross-sectional area. The outer shell carries an equal current in opposite direction and distributed uniformly. What will be the ratio of the magnetic field at a distance x from the axis when (i) x < a and (ii) a < x < b?
(1) \(\frac{x^2}{a^2}\)
(2) \(\frac{a^2}{x^2}\)
(3) \(\frac{x^2}{b^2 - a^2}\)
(4) \(\frac{b^2 - a^2}{x^2}\)
Correct answer: (1) \(\frac{x^2}{a^2}\)
Explanation:
when x < a
B1(2πx) = \(μ_o\Big(\frac{i_o}{\pi a^2}\Big)\pi x^2\)
B1(2πx) = \(\frac{μ_oi_o x^2}{a^2}\)
B1 = \(\frac{μ_oi_o x}{2 \pi a^2}\) .....(1)
when a < x < b
B2(2πx) = μoio
B1(2πx) = \(\frac{μ_oi_o}{2 \pi x}\) ......(2)
\(\frac{B_1}{B_2}\) = \(\frac{μ_oi_o \frac{x}{2 \pi a^2}}{\frac{μ_oi_o}{2 \pi x}}\)
= \(\frac{x^2}{a^2}\)