Question

The end points of a uniform conducting rod slide on the two mutually perpendicular non-conducting surfaces in a uniform magnetic field as shown. If velocity of end A is $V_0$ and the rod makes an angle of $53^\circ$ with the horizontal as shown.

Then the maximum value of $|V_A - V_P|$ is (where P is a point on the rod) ($V_A$ and $V_P$ are potential of point A and P respectively).

• A. $\frac{9}{125} B_0 \ell v_0$
• B. $\frac{18}{125} B_0 \ell v_0$
• C. $\frac{16}{125} B_0 \ell v_0$
• D. $\frac{32}{125} B_0 \ell v_0$

Answer 1

Answer: (D)

$\frac{32}{125} B_0 \ell v_0$

Answer 2

To find the maximum value of $|V_A - V_P|$, where P is a point on the rod, we analyze the potential difference between points A and P. The rod slides on two mutually perpendicular non-conducting surfaces in a uniform magnetic field $B_0$. The velocity of end A is $V_0$ horizontally, and the rod makes an angle of $53^\circ$ with the horizontal.

Let the corner be the origin (0,0). A is at $(x_A, 0)$, B is at $(0, y_B)$. The length of the rod is $\ell$. The angle with the horizontal is $53^\circ$. The instantaneous center of rotation (ICR) is at $(x_A, y_B)$.

The potential difference $V_A - V_P$ is given by: $V_A - V_P = -B_0 V_0 \left( \frac{s^2}{2\ell \sin\theta} - s\sin\theta \right)$

where $s$ is the distance from A to P.

To find the maximum value of $|V_A - V_P|$, we analyze the function $h(s) = \frac{s^2}{2\ell \sin\theta} - s\sin\theta$.

The minimum of $h(s)$ occurs at $s = \ell \sin^2\theta = \ell (4/5)^2 = \frac{16}{25}\ell$.

The minimum value is $h(\frac{16}{25}\ell) = -\frac{32}{125}\ell$.

The values of $V_A - V_P$ are: At $s=0$ (point P is A): $V_A - V_A = 0$. At $s=\ell$ (point P is B): $V_A - V_B = -B_0 V_0 \ell (\frac{1}{2\sin\theta} - \sin\theta) = \frac{7}{40} B_0 V_0 \ell$. At $s=\frac{16}{25}\ell$ (the minimum of $h(s)$): $V_A - V_P = \frac{32}{125} B_0 V_0 \ell$.

We need the maximum value of $|V_A - V_P|$. Comparing the absolute values of these results, we find that the maximum absolute value is $\frac{32}{125} B_0 V_0 \ell$.