Animated Solution for Physics - Electromagnetic Induction: A metal conductor of length $1\text{ m}$ rotates vertically about one of its ends at angular velocity $5\text{ rad/s}$. If the horizontal component of earth's magnetic field is $0.2\times 10^{-4}\text{ T}$, then the emf developed between the two ends of the conductor is

Imagine a metal rod acting like the sweeping hand of a giant clock, rotating steadily in a vertical plane. As it spins, it isn't just moving through empty space; it is slicing through the Earth's magnetic field. Specifically, because the rod is rotating vertically, it cuts perpendicularly across the horizontal component of the Earth's magnetic field ($B_H$).

This continuous "cutting" of magnetic field lines is the perfect recipe for electromagnetic induction. According to Faraday's Law and the Lorentz force acting on the free electrons inside the moving metal, an electromotive force (EMF) is generated across the length of the rod.

When a straight conductor of length $L$ rotates about one of its ends with an angular velocity $\omega$ in a uniform magnetic field $B$ that is perpendicular to the plane of rotation, the velocity of the rod isn't uniform. The pivot point is stationary, while the far tip is moving the fastest ($v=\omega L$).

To find the total induced EMF, we integrate the motional EMF ($de=Bvdx$) over the entire length of the rod. This elegant integration yields our master formula:

This equation is a high-yield concept in physics, elegantly linking rotational kinematics with electromagnetism.

Now, let's bring in the specific values provided in the problem. We are given the horizontal magnetic field $B=0.2\times 10^{-4}\text{ T}$, the angular velocity $\omega =5\text{ rad/s}$, and the length of the rod $L=1\text{ m}$. Substituting these into our master equation gives:

Let's break down the arithmetic. Half of $0.2$ is $0.1$, and $1^2$ is simply $1$. The expression simplifies beautifully:

To match the standard scientific notation and the given multiple-choice options, we need to adjust the decimal point. Moving the decimal two places to the right changes the power of ten from $-4$ to $-6$:

Since $10^{-6}\text{ V}$ is defined as one microvolt ($\mu \text{V}$), we arrive at our final, elegant result:

This tiny voltage is a direct consequence of the Earth's relatively weak magnetic field, but the physics governing it is universally powerful!