Induced EMF in a Rotating Metallic Rod

This problem involves understanding the concept of motional electromotive force (EMF) induced in a metallic rod rotating in a uniform magnetic field. The key is to correctly apply the formula for EMF induced in such a setup.

Understanding the Problem

We are given a scenario where a metallic rod of a specific length is rotated at a certain angular frequency about one of its ends. This rotation happens in a uniform magnetic field that is parallel to the axis of rotation. Our goal is to determine the electromotive force (EMF) induced between the two ends of this rotating rod.

Given Parameters

• Length of the metallic rod, \(L = 2.0 \, \text{m}\)
• Angular frequency of rotation, \(\omega = 100 \, \text{rad/s}\)
• Uniform magnetic field strength, \(B = 2.0 \, \text{T}\)
• The rod rotates about an axis normal to the rod and passing through one end.
• The magnetic field is parallel to the axis of rotation.

Motional EMF in a Rotating Rod

When a metallic rod rotates in a magnetic field, the free electrons within the rod experience a magnetic force due to their motion. This force causes the electrons to accumulate at one end of the rod, creating a potential difference across its ends. This potential difference is what we call the induced EMF.

For a rod of length \(L\) rotating with angular frequency \(\omega\) about one of its ends in a uniform magnetic field \(B\) perpendicular to the plane of rotation, the induced EMF (\(\varepsilon\)) is given by the formula:

\[ \varepsilon = \frac{1}{2} B \omega L^2 \]

It's important to note that the magnetic field is parallel to the axis of rotation, which means it is perpendicular to the velocity component of each segment of the rod as it moves. Hence, the standard formula for motional EMF applies.

Calculation of Induced EMF

Now, let's substitute the given values into the formula to calculate the induced EMF:

• Magnetic field, \(B = 2.0 \, \text{T}\)
• Angular frequency, \(\omega = 100 \, \text{rad/s}\)
• Length of the rod, \(L = 2.0 \, \text{m}\)

Substitute these values into the formula:

\[ \varepsilon = \frac{1}{2} \times (2.0 \, \text{T}) \times (100 \, \text{rad/s}) \times (2.0 \, \text{m})^2 \]

First, calculate the square of the length:

\[ L^2 = (2.0)^2 \, \text{m}^2 = 4.0 \, \text{m}^2 \]

Now, substitute this back into the EMF equation:

\[ \varepsilon = \frac{1}{2} \times 2.0 \times 100 \times 4.0 \]

Perform the multiplication:

\[ \varepsilon = 1.0 \times 100 \times 4.0 \]

\[ \varepsilon = 400 \, \text{V} \]

Therefore, the induced EMF between the two ends of the rod is 400 V.

Conclusion

The EMF induced between the two ends of the rotating rod is 400 V. This result is obtained by applying the formula for motional EMF in a rotating conductor, considering the given length, angular frequency, and magnetic field strength.