A rectangular coil of N turns, area A is held in a uniform magnetic field B. If the coil is rotated at a steady angular speed w, deduce an expression for the induced emf in coil at any time.

To derive the expression for the induced electromotive force (emf) in a rectangular coil that is rotating in a uniform magnetic field, we can start by applying Faraday's law of electromagnetic induction. This law states that the induced emf in a coil is directly related to the rate of change of magnetic flux through the coil. Let's break this down step by step.

Understanding Magnetic Flux

Magnetic flux (Φ) through a surface is defined as the product of the magnetic field (B) and the area (A) of the surface, adjusted for the angle (θ) between the magnetic field lines and the normal (perpendicular) to the surface. Mathematically, it can be expressed as:

Φ = B * A * cos(θ)

Setting Up the Problem

In our scenario, the coil has N turns and is rotating at a steady angular speed (ω) in a uniform magnetic field (B). As the coil rotates, the angle θ changes with time. If we denote the angular position of the coil at time t as θ(t) = ωt, then the magnetic flux through the coil can be expressed as:

Φ(t) = B * A * cos(ωt)

Calculating the Induced EMF

According to Faraday's law, the induced emf (ε) is the negative rate of change of magnetic flux through the coil:

ε = -dΦ/dt

Now, let's differentiate the magnetic flux with respect to time:

ε = -d/dt (B * A * cos(ωt))

Since B and A are constants, we can factor them out:

ε = -B * A * d/dt(cos(ωt))

Applying the Derivative

The derivative of cos(ωt) is -ωsin(ωt), so we substitute that in:

ε = -B * A * (-ωsin(ωt))

Thus, we simplify the expression to:

ε = B * A * ω * sin(ωt)

Final Expression for Induced EMF

The final expression for the induced emf in the rectangular coil as it rotates in the magnetic field at any time t is:

ε = B * A * ω * sin(ωt)

Understanding the Result

This expression highlights a few key points:

• The induced emf is proportional to the magnetic field strength (B), the area of the coil (A), and the angular speed (ω).
• The sine function indicates that the induced emf varies with time, reaching its maximum value when sin(ωt) = 1, which occurs at odd multiples of π/2.
• As the coil rotates, the direction of the induced emf will also reverse, which is a characteristic of alternating current (AC) systems.

In summary, the induced emf generated in a rotating coil is a dynamic process influenced by the coil's orientation in the magnetic field, and this relationship is crucial in various applications such as electric generators and motors. Understanding this principle allows you to appreciate how mechanical energy can be converted into electrical energy, which is fundamental to many technologies we use today.