Question

A square conducting loop with sides of length \(L\) is rotating at a constant angular speed, \(\omega\), in a uniform magnetic field of magnitude \(B\). At time \(t=0\), the loop is oriented so that the direction normal to the loop is aligned with the magnetic field. Find an expression for the potential difference induced in the loop as a function of time.

Short answer

Answer: The potential difference induced in the loop as a function of time is given by the expression Potential Difference(t) = B * L^2 * ω * sin(ω*t), where B is the magnetic field, L is the side length of the square loop, ω is the angular speed, and t is the time reference.

Step-by-step solution

Calculate the magnetic flux as a function of time

To calculate the magnetic flux through the loop, we must consider the geometry of the problem. The magnetic flux, Φ, is given by the dot product of the magnetic field, B, and the area vector, A: Φ = B · A However, as the loop rotates, the angle between the two vectors changes. We can express this angle as: θ(t) = ω * t Now we can find the magnetic flux through the loop in terms of time as: Φ(t) = B * A * cos(θ(t)) Given that A = L^2 (area of the square loop) and θ(t) = ω * t, the magnetic flux through the loop becomes: Φ(t) = B * L^2 * cos(ω*t)

Apply Faraday's Law to find the induced EMF

Faraday's Law states that the induced EMF in a loop is equal to the negative rate of change of the magnetic flux through the loop: EMF = -dΦ/dt To find the induced EMF, we differentiate the magnetic flux function we found in Step 1 with respect to time t: EMF(t) = -d(B * L^2 * cos(ω*t))/dt

Calculate the rate of change of magnetic flux

To calculate the derivative of the magnetic flux function, we apply the chain rule, while treating B and L^2 as constants: EMF(t) = -B * L^2 * d(cos(ω*t))/dt * d(ω*t)/dt Here, the derivative of ω*t with respect to t is just ω. The derivative of cos(ω*t) with respect to ω*t is -sin(ω*t). So the EMF function becomes: EMF(t) = -B * L^2 * (-sin(ω*t)) * ω

Simplify the EMF expression

Now, we simplify the EMF function by removing the double negative: EMF(t) = B * L^2 * ω * sin(ω*t)

Relate the induced EMF to the potential difference

In a conducting loop, the induced EMF is equal to the potential difference. Therefore, the expression for the potential difference induced in the loop as a function of time is: Potential Difference(t) = B * L^2 * ω * sin(ω*t)

Key concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle that explains how electric current can be generated by a changing magnetic environment. According to this law, an electromotive force (EMF) is induced in a closed circuit when the magnetic flux through the loop changes over time. The law is mathematically expressed as
\[ EMF = -\frac{d\Phi}{dt} \]
where \( EMF \) represents the induced electromotive force and \( d\Phi/dt \) denotes the rate of change of magnetic flux \( \Phi \) over time \( t \). The negative sign indicates the direction of the induced EMF as described by Lenz's law, which states that the induced EMF will generate a current whose magnetic field opposes the change in the original magnetic flux. This law is the principle behind many electrical transformers, generators, and motors, allowing for the conversion of mechanical movement into electrical energy.

Magnetic Flux

Magnetic flux, symbolized by \( \Phi \), is a measurement of the total magnetic field \( B \) passing through a given area \( A \). It is defined by the equation
\[ \Phi = B \cdot A \cos(\theta) \]
where \( \theta \) is the angle between the magnetic field and the area's normal direction. In the context of our rotating loop problem, as the loop turns, this angle changes with time, altering the flux through the loop.

Understanding magnetic flux is crucial for grasping how changes in a magnetic field can induce an EMF in a conductor. For instance, the quicker the loop rotates, or the stronger the magnetic field, the greater the rate of change of magnetic flux, which, according to Faraday's Law, leads to a larger induced EMF.

Uniform Magnetic Field

A uniform magnetic field is a field in which the magnetic field strength \( B \) and direction are consistent at all points across the area of interest. This consistency is essential for the rotating loop problem because it simplifies the calculation of magnetic flux and the subsequent determination of induced EMF. In a uniform magnetic field, any changes to the flux within our loop are solely due to the loop's rotation, not changes in the field strength or direction.

Visualize this as a uniform pattern of straight parallel lines representing the field, which ensures that no matter where the loop is within this field, the magnetic field interaction remains constant - only the orientation or the position of the loop would affect the magnetic flux.

Angular Velocity

Angular velocity, denoted by \( \omega \), is a vector quantity that specifies the rotational speed and direction of an object. It is especially relevant in our exercise as it denotes how quickly the square loop is spinning.

Mathematically, angular velocity is calculated by:
\[ \omega = \frac{d\theta}{dt} \]
where \( \theta \) is the rotational angle. In the case of our loop rotating in a magnetic field, the angular velocity affects how rapidly the magnetic field lines are being cut by the loop, which directly impacts the rate of change of magnetic flux and therefore the strength of the induced EMF. A faster rotation (higher \( \omega \)) leads to a quicker change in flux and a greater induced EMF.