Question

Magnetic Light Switch Ranking
Six identical vertical metal bars start at the positions shown below and move at constant velocities through identical magnetic fields. The bars make electrical contact with and move along frictionless metal rods attached to light bulbs.

Part A

At the instant shown, rank these six scenarios on the basis of the magnitude of the current in the lightbulb. Rank from largest to smallest. To rank items as equivalent, overlap them.

Answer 1

Concepts and reason

The concept required to solve the given problem is induced emf.

First the expression for the induced emf in the vertical bar is calculated and then the relationship between the current and velocity of the vertical bar will be established.

Fundamentals

Induced emf: It is defined as the rate of change of magnetic flux linked with a coil.

$\varepsilon = Blv$

Here, B is the magnitude of magnetic field, l is the length of the coil, and v is the velocity.

According to Ohm’s law, current through a conductor is given as,

$I = \frac{\varepsilon }{R}$

Here, R is the resistance of the conductor.

(A)

The expression for the induced emf in terms of motional emf in the vertical bar is,

$\varepsilon = Blv$

Here, B is the magnitude of magnetic field, l is the length of the vertical bar, and v is the velocity of the bar.

The expression for the current in the light bulb is,

$i = \frac{\varepsilon }{r}$

Here, i is the current in the light bulb and r is the resistance of the blub.

Substitute $Blv$ for $\varepsilon$ in equation $i = \frac{\varepsilon }{r}$ .

$\begin{array}{c}\\i = \frac{\varepsilon }{r}\\\\ = \frac{{Blv}}{r}\\\end{array}$

Therefore, the magnitude of current is directly proportional to the velocity of the bar. i.e.,

$\begin{array}{l}\\i \propto v\\\\i = kv\\\end{array}$

Here, k is the constant.

The current through the light bulb is proportional to the velocity of the bar.

Item: 1: Substitute 10 cm/s for v in equation $i = kv$ .

$i = 10k{\rm{ cm/s}}$

Item: 2: Substitute 20 cm/s for v in equation as follows:

$i = 20k{\rm{ cm/s}}$

Item: 3: Substitute 5 cm/s for v in equation as follows:

$i = 5k{\rm{ cm/s}}$

Item: 4: Substitute 0 cm/s for v in equation as follows:

$\begin{array}{c}\\i = \left( {0{\rm{ cm/s}}} \right)k\\\\ = 0{\rm{ }}\\\end{array}$

Ans: Part A

Ranking from the largest to the smallest current (in terms of velocities) is,

Answer 2

ref:  https://www.homeworklib.com/question/942470/six-identical-vertical-metal-bars-start-at-the