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 $Y$
$\Rightarrow U_{X}=U_{Y}$
$\Rightarrow \frac{B_{X}^{2} V_{X}}{2 \mu_{0}}=\frac{B_{Y}^{2} V_{Y}}{2 \mu_{0}}$
$\left[\because U=\frac{B^{2} V}{2 \mu_{0}}, V=\text { volume }\right]$
$B_{X}^{2} V_{X}=B_{Y}{ }^{2} V_{Y}$
$\Rightarrow B_{X}{ }^{2} A_{X} L_{X}=B_{Y}{ }^{2} A_{Y} L_{Y}$
$\Rightarrow \frac{B_{X}{ }^{2}}{B_{Y}{ }^{2}}=\frac{A_{Y}}{A_{X}} \times \frac{L_{Y}}{L_{X}}=\frac{2 A_{X}}{A_{X}} \times \frac{2 L_{X}}{L_{X}}$
$\Rightarrow \frac{B_{X}{ }^{2}}{B_{Y}{ }^{2}}=4$
$\Rightarrow \frac{B_{X}}{B_{Y}}=\sqrt{4}$
$\Rightarrow \frac{B_{X}}{B_{Y}}=2$
$\Rightarrow B_{X}: B_{Y}=2: 1$