when a rod is pivoted at centre and made to rotate in a magnetic field find the net emf induced

To determine the net electromotive force (emf) induced in a rod that is pivoted at its center and rotates in a magnetic field, we can apply Faraday's law of electromagnetic induction. This principle states that the induced emf in a closed loop is equal to the rate of change of magnetic flux through the loop. Let’s break this down step by step.

Understanding the Setup

Imagine a straight rod of length L that is pivoted at its center. When this rod rotates in a uniform magnetic field, the motion of the rod through the magnetic field lines induces an emf. The magnetic field can be represented as being uniform and directed perpendicular to the plane of rotation.

Key Concepts

• Faraday's Law: The induced emf (ε) is given by the formula ε = -dΦ/dt, where Φ is the magnetic flux.
• Magnetic Flux (Φ): This is defined as Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area through which the field lines pass, and θ is the angle between the magnetic field and the normal to the surface area.
• Length of the Rod: The rod rotates about its center, meaning each half of the rod will have a different linear velocity as it moves through the magnetic field.

Calculating the Induced EMF

Let’s denote the magnetic field strength as B, the length of the rod as L, and the angular velocity of the rod as ω (in radians per second). The key to finding the induced emf is to consider the motion of each segment of the rod.

Finding the Area Swept by the Rod

As the rod rotates, the distance from the pivot to any point along the rod varies. The total length of the rod is L, so each half of the rod (length L/2) sweeps out a circular area as it rotates. The area A swept out by the rod in a time interval dt can be expressed as:

A = (L/2) * (r * dt), where r is the distance from the pivot to the end of the rod.

Induced EMF Calculation

The induced emf can be calculated by integrating the contributions from each differential segment of the rod. The emf induced in a small segment of the rod, dL, at a distance r from the pivot is given by:

dε = B * v * dL, where v = r * ω (the linear velocity of the segment).

Substituting this into the equation gives:

dε = B * (r * ω) * dL.

Integrating Over the Length of the Rod

To find the total emf, we integrate from 0 to L/2 (since we only need to consider half the rod due to symmetry):

ε = ∫(B * (r * ω) * dL) from 0 to L/2.

After performing the integration, we find:

ε = (B * ω * (L^2/8)), which simplifies to ε = (B * ω * L^2) / 8.

Final Thoughts

The net emf induced in the rod rotating in a magnetic field is directly proportional to the magnetic field strength, the angular velocity of the rod, and the square of the length of the rod. This relationship highlights how motion in a magnetic field can generate electrical energy, a principle that underpins many technologies, including electric generators.