Magnetic Effects & Magnetism

Olympiad (NSEP) level, single correct, hard

3 of 276

Question 3 of 276

A coaxial cable consists of an inner solid wire of radius aa and an outer thin cylindrical shell of radius bb carrying equal and opposite steady current II. The magnetic energy stored per unit length in the space between aa and bb is:

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Answer: (B) μ0I24πln⁡(b/a)\frac{\mu_0 I^2}{4\pi} \ln(b/a)

Between conductors, B(r)=μ0I2πrB(r) = \frac{\mu_0 I}{2\pi r}. Energy density uB=B22μ0=μ0I28π2r2u_B = \frac{B^2}{2\mu_0} = \frac{\mu_0 I^2}{8\pi^2 r^2}. Integrating over volume: U/L=∫abuB2πrdr=μ0I24π∫abdrr=μ0I24πln⁡(b/a)U/L = \int_a^b u_B 2\pi r dr = \frac{\mu_0 I^2}{4\pi} \int_a^b \frac{dr}{r} = \frac{\mu_0 I^2}{4\pi} \ln(b/a).
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