Question

A ring of radius \(\mathrm{r}\) is rotating about its diameter with angular velocity w in a perpendicular magnetic field \(\mathrm{B}^{-}\) It has 20 turns. The emf induced is (a) \(20 \mathrm{~B} \pi \mathrm{r}^{2} \sin \mathrm{wt}\) (b) \(20 \mathrm{~B} \pi \mathrm{r}^{2} \mathrm{cos} \mathrm{wt}\) (c) \(10 \sqrt{2} \mathrm{~B} \pi \mathrm{r}^{2}\) (d) \(20 \mathrm{~B} \pi \mathrm{r}^{2} \mathrm{w} \sin \mathrm{wt}\)

Short answer

The induced emf in the rotating ring is given by the expression \(emf = 20 B \pi r^2 w \sin(wt)\), so the correct answer is (a) \(20 B \pi r^{2} \sin(wt)\).

Step-by-step solution

Recall Faraday's law of electromagnetic induction

According to Faraday's law, the induced emf in a loop is proportional to the rate of change of magnetic flux through the loop. Mathematically, it can be expressed as: \(emf = -N \frac{dΦ}{dt}\) where \(emf\) is the induced emf, \(N\) is the number of turns, and \(\frac{dΦ}{dt}\) is the rate of change of magnetic flux.

Calculate the magnetic flux through the ring

The magnetic flux, Φ, through a surface is given by the dot product of the magnetic field, B, and the area vector through which the magnetic field lines are passing. Since the magnetic field is perpendicular to the plane of the ring, the magnetic flux can be expressed as: Φ = BAcosθ where A is the area of the ring, and θ is the angle between the magnetic field and the area vector. Since the ring is rotating about its diameter, we can write θ as a function of time: θ(t) = w*t where w is the angular velocity and t is the time.

Compute the rate of change of magnetic flux

Now, we need to find the rate of change of magnetic flux with respect to time. To do this, we take the derivative of Φ(t) with respect to t: \(\frac{dΦ}{dt} = \frac{d}{dt}( BAcos(wt))\) We can factor out B and A as they are constants: \(\frac{dΦ}{dt} = BA\frac{d}{dt}(cos(wt))\) Now, differentiate cos(wt) with respect to t: \(\frac{dΦ}{dt} = BA(-w \sin(wt))\) So, \(\frac{dΦ}{dt} = -BAw \sin(wt)\)

Calculate the induced emf

Now, we can use Faraday's law to calculate the induced emf: \(emf = -N \frac{dΦ}{dt}\) We have N = 20 turns, and we found the rate of change of magnetic flux in Step 3. Substituting the values, we get: \(emf = 20(-(-BAw \sin(wt)))\) Finally, we can simplify the expression: \(emf = 20 BAw \sin(wt)\) Recall that A = πr², where r is the radius of the ring. Plugging this into the equation, we get: \(emf = 20 B \pi r^2 w \sin(wt)\) So, the correct answer is (a) \(20 B \pi r^{2} \sin(wt)\).

Key concepts

Faraday's Law

Faraday's Law of electromagnetic induction is a fundamental principle that describes how electric currents are generated within conductive loops when they are exposed to changing magnetic fields. This law helps explain the operation of many electrical devices, including motors and generators.
Faraday reasoned that changing magnetic environments induce electric fields, leading to an electromotive force (EMF). In simple terms:

• EMF is generated proportional to the rate of change of magnetic flux through the loop.
• The formula is: \( emf = -N \frac{d\Phi}{dt} \)
• \(N\) stands for the number of turns in the coil.
• The negative sign incorporates Lenz's law, indicating that the induced current will oppose the change in magnetic flux.

Understanding Faraday's Law is essential for comprehending how electricity is produced in rotating loops affected by magnetic fields.

Magnetic Flux

Magnetic Flux, represented by the symbol \(\Phi\), describes the amount of magnetic field passing through a surface such as a loop of wire. It helps in determining how a magnetic field interacts with different objects.
The calculation of magnetic flux is straightforward:

• It’s given by the formula: \(\Phi = BA \cos \theta\).
• \(B\) is the magnetic field strength.
• \(A\) is the area through which the field lines pass.
• \(\theta\) is the angle between the magnetic field and the perpendicular to the area.

For our rotating ring exercise, \(\theta\) changes with time due to its rotation, hence \(\Phi\) becomes a function of time. This time dependency is key in inducing EMF.

Induced EMF

Induced EMF (Electromotive Force) arises when a change occurs in the magnetic environment around a conductor. This changing magnetic environment can be due to a moving magnet, a changing magnetic field, or as in this exercise, the rotation of the loop itself.
Based on Faraday's Law, the magnitude of the induced EMF is:

• \(emf = -N \frac{d\Phi}{dt}\), where the rate of change of \(\Phi\) plays a crucial role.
• As the flux changes, an EMF circular in the loop gets created.
• It causes current flow if a closed path is present.

In practice, this concept forms the basis of many practical applications, from electric generators to transformers, contributing to energy conversion processes extensively.

Angular Velocity

Angular Velocity, denoted by \(\omega\), measures how fast an object rotates over a certain time period around an axis. It is a vector quantity, indicating both the speed of rotation and the direction of the axis.
In the context of our exercise, angular velocity:

• Relates directly to the rate of flux change through the rotating loop.
• Expresses in radians per second \(\text{(rad/sec)}\).
• Influences the angle \(\theta(t) = \omega t\) in the magnetic flux formula \(\Phi = BA\cos(\theta(t))\).

Understanding the role of angular velocity is critical in solving problems dealing with rotating systems in magnetic fields, as it directly impacts both the magnitude and frequency of the induced EMF.