Question

Consider the circuit shown in Fig., but with the bar moving to the right with speed v. As in Exercise 31, the bar has a length of 0.360 m, R= 45.0, and B= 0.650 T.

(a) Is the induced current in the circuit clockwise or counterclockwise?

(b) At an instant when the 45.0$\mathbf{\Omega }$resistor is dissipating electrical energy at a rate of 0.840 J/s, what is the speed of the bar?

Short answer

1. The induced current in the circuit clockwise direction.
2. At an instant when the 45.0 resistor is dissipating electrical energy at a rate of 0.840 J/s, the speed of the bar is 26.3 m/s.

Step-by-step solution

Given

We have a conducting rod ab, which makes contact with metal rails ca and db where the parallel metal rails are connected through a R = 45.0resistor, the whole device is placed perpendicularity in a magnetic field of B = 0.650 T, as shown in the following figure.

Calculate the direction of the induced current in the circuit.

We need to find the direction of induced current in the circuit when the rod is moving under influence of an applied force F toward the left at the instant when the speed is v= 5.90 m/s as shown in the figure. Let x be the length of the expanding side db, and L = 0.360 m is the length of the constant length side ab. The area of the loop abcd decreases as the bar moves to the left, hence the magnetic flux and the external magnetic field point out of the page, so the induced magnetic field must point out of the page (according to the Lenz's law), so the induced current must circulate counterclockwise in the circuit.

Calculate the speed of the bar.

The induced current in the resistor equals the induced emf divided by the resistance, that is,

$\mathrm{I}=\frac{\mathrm{\epsilon }}{\mathrm{R}}$

where,

$\mathrm{\epsilon }=\mathrm{vBL}$

so,

$\mathrm{I}=\frac{\mathrm{vBL}}{\mathrm{R}}$

the rate at which the resistor dissipates electrical energy is,

$P_R=I^{2}R$

Substitute with we get,

${\mathrm{P}}_{\mathrm{R}}=\frac{{\mathrm{v}}^2{\mathrm{B}}^2\mathrm{L}}{\mathrm{R}}$

we need to find the speed of the bar at an instant when the resistor is dissipating electrical energy at a rate of ${\mathrm{P}}_{\mathrm{R}}=0.840\mathrm{J}/\mathrm{s}$= 0.840J /s substitute with the givens we get

$$\begin{gathered} v=\sqrt{\frac{P_{R}R}{B^{2}L}} \\ =\sqrt{\frac{\left(.840J/s\right)\left(45.0\Omega \right)}{{\left(0.650T\right)}^2{\left(0.360m\right)}^2}} \\ =26.3m/s \\ v=26.3m/s \end{gathered}$$

Figure.