A metal rod of length 0.4\text{ m} glides at a constant speed of 5.0\text{ m s}^{-1} to th… — JAMB UTME

A metal rod of length $0.4\text{ m}$ glides at a constant speed of $5.0\text{ m s}^{-1}$ to the right along horizontal parallel conducting rails placed in a uniform magnetic field of $0.5\text{ T}$ directed vertically into the page. The rails are connected at their left end by a $2.0\ \Omega$ resistor. What is the magnitude of the external force required to maintain the uniform speed of the rod, and in which direction does the induced magnetic force act on the rod?

$0.10\text{ N}$ , directed to the leftAnswer
B
$0.10\text{ N}$ , directed to the right
C
$0.50\text{ N}$ , directed to the left
D
$0.50\text{ N}$ , directed to the right

Answer

0.10\text{ N}

, directed to the left

The motional e.m.f. generated across the moving conductor is

\mathcal{E} = BLv = 0.5 \times 0.4 \times 5.0 = 1.0\text{ V}

. The induced current is

I = \mathcal{E}/R = 1.0 / 2.0 = 0.5\text{ A}

. The magnetic force resisting the rod's motion is

F_B = BIL = 0.5 \times 0.5 \times 0.4 = 0.10\text{ N}

. By Lenz's law, this magnetic force acts to the left, opposing the motion to the right. To maintain uniform speed, an equal external force of

0.10\text{ N}

must be applied.

Step-by-Step Solution

Calculate the induced electromotive force (e.m.f.) across the moving rod

\mathcal{E} = B L v = 0.5\text{ T} \times 0.4\text{ m} \times 5.0\text{ m s}^{-1} = 1.0\text{ V}

Motional e.m.f. is produced when a conductor cuts magnetic flux lines at a perpendicular velocity.

Calculate the induced current flowing in the circuit

I = \frac{\mathcal{E}}{R} = \frac{1.0\text{ V}}{2.0\ \Omega} = 0.5\text{ A}

Ohm's law relates the induced e.m.f. and total circuit resistance.

Determine the magnitude of the magnetic force acting on the current-carrying rod

F_B = B I L = 0.5\text{ T} \times 0.5\text{ A} \times 0.4\text{ m} = 0.10\text{ N}

A magnetic field exerts a force on a straight conductor carrying current.

Determine the direction of the magnetic force using Lenz's law and Newton's first law

The magnetic force acts to the left (opposing motion to the right). An external force of equal magnitude (

0.10\text{ N}

) to the right is required to maintain constant speed.

Lenz's law states that induced effects always oppose the change causing them (motion to the right).

Key Concept

Motional Electromotive Force, Magnetic Force on a Conductor, and Lenz's Law

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