Question

Currents i and 3i are flowing in opposite directions in two co-axial long thin walled circular cylinders of radii a and 2a respectively as shown. Choose correct options. (Length of the cylinder is L)

• A. Pressure on inner cylinder wall is zero.
• B. Pressure on inner cylinder wall is ${\mu }_0\frac{i^2}{8{\pi }^2{a}^2}$.
• C. Pressure on outer cylinder wall is $3{\mu }_0\frac{i^2}{32{\pi }^2{a}^2}$.
• D. Force on unit length of outer cylinder is $\frac{3}{4}\frac{{\mu }_0{i}^2}{\pi a}$.

Answer 1

Answer: (B), (C)

The correct options are
B Pressure on inner cylinder wall is ${\mu }_0\frac{i^2}{8{\pi }^2{a}^2}$.
C Pressure on outer cylinder wall is $3{\mu }_0\frac{i^2}{32{\pi }^2{a}^2}$.

Magnetic field at surface of inner cylinder is $B_1=\frac{{\mu }_0}{4\pi }\frac{i}{a}$ [By ampere's law]
Force on L length of cylinder $F=iL{B}_1$
$F=\frac{{\mu }_0{i}^{2}L}{4\pi a}$ & $P=\frac{F}{A}=\frac{{\mu }_0{i}^{2}L}{4\pi a\times 2\pi aL}=\frac{{\mu }_0{i}^2}{8{\pi }^2{a}^2}$
Field at surface of outer cylinder is
$B_2=\frac{{\mu }_0}{2\pi }\frac{i}{2a}-\frac{{\mu }_0}{4\pi }.\frac{3i}{2a}\Rightarrow |B_2|=\frac{{\mu }_{0}i}{8\pi a}$
$F=\frac{{\mu }_0}{8a\pi }3iL$ & $P=\frac{F}{A}=\frac{{\mu }_{0}i\times 3iL}{8\pi a\times 2\pi \times 2aL}$
$P=\frac{3}{32}\frac{{\mu }_0{i}^2}{{\pi }^2{a}^2}$