Q152 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Physics

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

  • A
    1 : 4
  • Correct answer
    B
    2 : 1
  • C
    1 : 2
  • D
    4 : 1

Solution

According to given condition, Magnetic energy stored in solenoid $X=$ Magnetic energy stored in solenoid $Y$ $\begin{array}{ll} \Rightarrow & U_X=U_Y \\ \Rightarrow & \frac{B_X^2 V_X}{2 \mu_0}=\frac{B_Y^2 V_Y}{2 \mu_0}\left[\because U=\frac{B^2 V}{2 \mu_0}, V=\text { volume }\right] \\ \Rightarrow \quad & B_X^2 V_X=B_Y^2 V_Y \\ \Rightarrow \quad & B_X^2 A_X L_X=B_Y^2 A_Y L_Y \\ \Rightarrow \quad & \frac{B_X^2}{B_Y^2}=\frac{A_Y}{A_X} \times \frac{L_Y}{L_X}=\frac{2 A_X}{A_X} \times \frac{2 L_X}{L_X} \\ \Rightarrow \quad & \frac{B_X^2}{B_Y^2}=4 \Rightarrow \frac{B_X}{B_Y}=\sqrt{4} \\ & \frac{B_X}{B_Y}=2 \Rightarrow B_X: B_Y=2: 1 \end{array}$

Hence, the correct answer is option: 'B'.

Relevant Questions