Question

Consider the arrangement shown in the figure below. Assume R = 7.0 Ω and = 1.2 m, and a uniform 3.0 T magnetic field is directed into the page. At what speed should the bar be moved to produce a current of 0.50 A in the resistor?

m/s?

Answer 1

Concepts and reason

The concepts used in this problem are magnetic flux, Ohm’s law and induced emf.

Initially, derive the expression for the speed of the bar using the equation of magnetic flux and induced emf. Then substitute the values in the expression of speed to calculate the speed of the bar.

Fundamentals

Magnetic Flux

The magnetic flux is the number of field lines passing through the area or surface of a conductor. The expression for magnetic flux passing through a loop having only one turn is,

Here, is the magnetic flux, is the magnetic field and is the area and

Electromotive force (emf)

The rate of change of magnetic flux is called emf. The expression for emf is,

Here, is the induced emf, is the rate of change of flux.

Ohm’s Law:

According to Ohm’s Law, “the voltage is directly proportional to the current”

Here, is the voltage, is the current and is the resistance.

The area of the rectangle is,

…… (1)

Here, is the area, is the length and is the width.

The magnetic flux is,

…… (2)

Substitute equation (2) in equation (1).

…… (3)

The emf generated in the conductor is,

…… (4)

Substitute equation (3) in equation (4).

The bar is moving horizontally, so the width of the loop changes with time.

Substitute for in the above expression.

Here, is the speed.

The expression of emf is,

…… (5)

The expression of emf in terms of current and resistance is,

…… (6)

Compare equation (5) and equation (6).

…… (7)

Substitute for, for, for and for in the equation (7).

The negative sign indicates the direction of movement of the bar. The speed is a scalar quantity, so direction is not required.

Ans:

The speed of the bar is.