Question

A straight metallic rod of length $L$ is oriented horizontally and dropped vertically downward with a speed $v$ perpendicular to the horizontal component of the Earth's uniform magnetic field $B_H$. The instantaneous motional electromotive force (EMF) induced across the two ends of this rod is:

• A. \( \frac{1}{2}B_H v L^2 \)
• B. \( B_H v L \)
• C. \( \frac{B_H v}{L} \)
• D. \( 0 \)

Hint:

For motional EMF to exist, the moving conductor must actively cut across magnetic flux lines. If you align the rod along the North-South direction and drop it, its length axis runs parallel to the field lines, which means zero lines are cut and the induced EMF drops to exactly zero.

Answer 1

The Correct Option is B

Concept: Motional electromotive force (EMF) is an electrical potential difference induced across a conductor that is actively moving through a magnetic field region. At the microscopic level, as the metallic conductor moves through the field, the free conduction electrons inside the metal move along with it at velocity \(\vec{v}\). Because these charges are moving within an external magnetic field \(\vec{B}\), they experience a magnetic Lorentz force (\(\vec{F} = q(\vec{v} \times \vec{B})\)) that pushes them along the length of the conductor. This force drives free electrons toward one end of the rod, leaving behind a net positive charge at the opposite end. This separation of charges creates an internal electric field that opposes the magnetic force. Eventually, a dynamic equilibrium is reached, establishing a stable voltage difference across the ends of the conductor. The vector equation for this induced potential difference is written as: \[ e = (\vec{v} \times \vec{B}) \cdot \vec{L} \] When the velocity vector \(\vec{v}\), the magnetic field vector \(\vec{B}\), and the straight length axis vector \(\vec{L}\) are all mutually perpendicular to each other, this scalar triple product simplifies to: \[ e = B v L \]

Step 1: Analyze the geometric alignment of the three key vectors. Let's look at how the vectors in the problem are oriented relative to each other:
• Magnetic Field Vector (\(\vec{B}\)): Aligned with the horizontal component of Earth’s magnetic field (\(B_H\)). Let's say it points along the North-South direction.
• Velocity Vector (\(\vec{v}\)): The rod is dropped straight down, meaning its velocity vector points vertically downward toward the ground.
• Conductor Length Axis (\(\vec{L}\)): The rod is held horizontally, running along the East-West direction to cut across the field lines. This layout confirms that the vertical velocity path, the horizontal North-South field lines, and the horizontal East-West rod length are all perfectly perpendicular to one another (separated by an angle of \(90^\circ\)).

Step 2: Calculate the magnitude of the induced motional EMF. Since all three components are mutually orthogonal, the sine and cosine parameters from our vector cross and dot products both evaluate to a maximum value of 1: \[ e = B_H \cdot v \cdot L \cdot \sin(90^\circ) \cdot \cos(0^\circ) \] \[ e = B_H v L \] As a result, a stable potential difference of exactly \(B_H v L\) is generated across the tips of the falling rod.