Question

Consider two solenoids $X$ and $Y$ such that the area and length of $Y$ are twice that of $X$ respectively, of the magnetic energy stored in both the solenoids is same, then the ratio of magnitude of magnetic fields of the two solenoids $\left|\frac{\vec{B}x}{\vec{B}\gamma }\right|$ is

• A. $1:4$
• B. $2:1$
• C. $1:2$
• D. $4:1$

Answer 1

Answer: (B)

According to given condition, Magnetic energy stored in solenoid $X=$ Magnetic energy stored in solenoid $Y$
$\Rightarrow U_X=U_Y$
$\Rightarrow \frac{B_X^2{V}_X}{2{\mu }_0}=\frac{B_Y^2{V}_Y}{2{\mu }_0}$
$\left[\because U=\frac{B^{2}V}{2{\mu }_0},V=\text{ volume }\right]$
$B_X^2{V}_X=B_Y{}^2{V}_Y$
$\Rightarrow B_X{}^2{A}_X{L}_X=B_Y{}^2{A}_Y{L}_Y$
$\Rightarrow \frac{B_X{}^2}{B_Y{}^2}=\frac{A_Y}{A_X}\times \frac{L_Y}{L_X}=\frac{2A_X}{A_X}\times \frac{2L_X}{L_X}$
$\Rightarrow \frac{B_X{}^2}{B_Y{}^2}=4$
$\Rightarrow \frac{B_X}{B_Y}=\sqrt{4}$
$\Rightarrow \frac{B_X}{B_Y}=2$
$\Rightarrow B_X:B_Y=2:1$