a conducting circular loop of radius a is placed in a uniform ,perpendicular magnetic field B. A thick metal rod OA is pivoted at the centre O. The other end of the rod touches the loop at A.The centre O and fixed point C on the loop are connected by wire OC.A force is applied at the middle point of rod OA perpendicularly,so that the rod rotates clockwise at uniform angular velocity w.Resistance of OC zero.circular loop has resistanceR uniformly distributed on it.Find current in rod,angleAOC90

To solve this problem, we need to analyze the situation involving the conducting circular loop, the metal rod, and the magnetic field. The key points to consider are the induced electromotive force (emf) in the loop due to the motion of the rod and the resulting current. Let's break it down step by step.

Understanding the Setup

We have a circular loop of radius a placed in a uniform magnetic field B. A thick metal rod OA is pivoted at the center O of the loop, with the other end touching the loop at point A. The wire OC connects the center O to a fixed point C on the loop. When a force is applied to the midpoint of the rod, it rotates clockwise with a uniform angular velocity w.

Induced EMF Calculation

As the rod rotates, it cuts through the magnetic field lines, which induces an electromotive force (emf) in the loop. According to Faraday's law of electromagnetic induction, the induced emf (ε) can be calculated using the formula:

• ε = -dΦ/dt

Where Φ is the magnetic flux through the loop. The magnetic flux is given by:

• Φ = B * A

Here, A is the area of the loop that is effectively being swept by the rod as it rotates. The area can be expressed as:

• A = (1/2) * a^2 * θ

Where θ is the angle in radians that the rod has rotated. Since the rod rotates at a uniform angular velocity w, we can express θ as:

• θ = wt

Substituting this into the area formula gives:

• A = (1/2) * a^2 * (wt)

Now, substituting this into the flux equation:

• Φ = B * (1/2) * a^2 * (wt)

Taking the derivative with respect to time to find the induced emf:

• ε = -d/dt [B * (1/2) * a^2 * (wt)] = -B * (1/2) * a^2 * w

Current in the Loop

Now that we have the induced emf, we can find the current in the loop. The current (I) can be calculated using Ohm's law:

• I = ε/R

Substituting the expression for ε gives:

• I = (-B * (1/2) * a^2 * w) / R

Since we are interested in the magnitude of the current, we can drop the negative sign:

• I = (B * (1/2) * a^2 * w) / R

Current in the Rod

The current in the rod OA is the same as the current in the loop since they are connected. Therefore, the current flowing through the rod is:

• I = (B * (1/2) * a^2 * w) / R

Final Thoughts

In summary, the current in the rod OA, when it is rotated at a uniform angular velocity w in a perpendicular magnetic field B, is given by the formula:

• I = (B * (1/2) * a^2 * w) / R

This result illustrates the relationship between the motion of the rod, the magnetic field, and the induced current in the conducting loop. Understanding these principles is crucial in the study of electromagnetism and its applications in various technologies.