Question

A circular conducting loop with radius \(a\) and resistance \(R_{2}\) is concentric with a circular conducting loop with radius \(b \gg a(b\) much greater than \(a\) ) and resistance \(R_{1}\). A time-dependent voltage is applied to the larger loop, having a slow sinusoidal variation in time given by \(V(t)=V_{0} \sin \omega t\) where \(V_{0}\) and \(\omega\) are constants with dimensions of voltage and inverse time, respectively. Assuming that the magnetic field throughout the inner loop is uniform (constant in space) and equal to the field at the center of the loop, derive expressions for the potential difference induced in the inner loop and the current \(i\) through that loop.

Short answer

Answer: The final expressions for the potential difference induced in the inner loop and the current through that loop are: $$V_{induced}(t) = - \pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}}\right)\omega\cos\omega t$$ $$i(t) = -\pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}R_{2}}\right)\omega\cos\omega t$$

Step-by-step solution

Determine the current in the outer loop

The current \(I_{1}\) in the outer loop can be determined using Ohm's law. Ohm's law states that the voltage across a resistor is equal to the product of the resistance and the current through it. Thus, we have: $$I_{1}(t) = \frac{V(t)}{R_{1}} = \frac{V_{0} \sin \omega t}{R_{1}}$$

Calculate the magnetic field produced by the outer loop

The magnetic field produced by a circular current loop at its center can be obtained by applying Biot-Savart law or Ampere's law. Here we will use Ampere's law, which states that the integral of the magnetic field around a closed-loop path is equal to the total enclosed current multiplied by the permeability of free space, \(\mu_{0}\). For a circular loop with current \(I_{1}\) and radius \(b\), the magnetic field at the center is given by: $$B(t) = \mu_{0}\frac{I_{1}(t)}{2b} = \frac{\mu_{0} V_{0}}{2b R_{1}} \sin \omega t$$

Apply Faraday's law to derive the induced potential difference

Faraday's law of electromagnetic induction states that the induced electromotive force (EMF) in a closed loop is equal to the rate of change of the magnetic flux through the loop. For the inner circular loop with radius \(a\), the potential difference induced across it can be given by: $$V_{induced}(t) = - \frac{d\Phi}{dt}$$ where \(\Phi\) is the magnetic flux through the inner loop and the negative sign indicates Lenz's law. We know that the magnetic field throughout the inner loop is assumed to be uniform and equal to the field at the center of the loop. Therefore, the magnetic flux through the inner loop is: $$\Phi(t) = B(t)\pi a^2$$ Now, take the derivative of \(\Phi(t)\) with respect to \(t\): $$\frac{d\Phi}{dt} = \pi a^2 \frac{d}{dt}\left(\frac{\mu_{0} V_{0}}{2b R_{1}} \sin \omega t\right)$$ By applying the chain rule, we get: $$\frac{d\Phi}{dt} = \pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}}\right)\omega\cos\omega t$$ Therefore, the induced potential difference in the inner loop is given by: $$V_{induced}(t) = - \pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}}\right)\omega\cos\omega t$$

Calculate the current in the inner loop

To find the current \(i\) through the inner loop, we can use Ohm's law, which states that the voltage across the resistor is equal to the product of the resistance and the current through it. Thus, we have: $$i(t) = \frac{V_{induced}(t)}{R_{2}} = -\pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}R_{2}}\right)\omega\cos\omega t$$ The final expressions for the potential difference induced in the inner loop and the current through that loop are: $$V_{induced}(t) = - \pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}}\right)\omega\cos\omega t$$ $$i(t) = -\pi a^2 \left(\frac{\mu_{0} V_{0}}{2b R_{1}R_{2}}\right)\omega\cos\omega t$$

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering and physics. It states that the current (\(I\)) through a conductor between two points is directly proportional to the voltage (\(V\)) across the two points and inversely proportional to the resistance (\(R\)) of the conductor.
The mathematical expression for Ohm's Law is:

• \(V = IR\)

In the context of this exercise, Ohm's Law is used to determine the current in the outer loop. Given the sinusoidal input voltage \(V(t) = V_0 \sin \omega t\), the current \(I_1(t)\) is calculated by dividing the voltage by the resistance \(R_1\):

• \(I_1(t) = \frac{V_0 \sin \omega t}{R_1}\)

This equation shows how the current changes over time, following the same sinusoidal pattern as the applied voltage, due to the proportionality described by Ohm's Law. Understanding this relationship is crucial for analyzing circuits with alternating voltage sources.

Ampere's Law

Ampere's Law is a powerful tool in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It states that the integrated magnetic field (\(B\)) along a closed path is proportional to the enclosed current.
Mathematically, Ampere's Law is represented as:

• \(\oint B \cdot dl = \mu_0 I_{enclosed}\)

where \(\mu_0\) is the permeability of free space.
In the exercise, Ampere's Law helps calculate the magnetic field at the center of the outer circular loop. By assuming that the magnetic field is uniform across the loop, we derive:

• \(B(t) = \frac{\mu_0 I_1(t)}{2b} = \frac{\mu_0 V_0}{2b R_1} \sin \omega t\)

This relationship links the current in the loop to the magnetic field intensity, demonstrating how electric currents generate magnetic fields, essential for understanding electromagnetic systems.

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle that describes how electric currents can be induced by changing magnetic fields. It states that the electromotive force (EMF) induced in a loop is equal to the negative rate of change of magnetic flux through the loop.
In formula terms, it is expressed as:

• \(\mathcal{E} = - \frac{d\Phi}{dt}\)

where \(\Phi\) is the magnetic flux.
For this exercise, Faraday's Law is applied to the inner loop to determine the induced potential difference as the sinusoidal magnetic field changes over time. The flux through the loop is:

• \(\Phi(t) = B(t) \pi a^2\)

Taking the time derivative of \(\Phi(t)\) gives the induced EMF,

• \(V_{induced}(t) = - \pi a^2 \left(\frac{\mu_0 V_0}{2b R_1}\right) \omega \cos \omega t\)

This expression highlights how the changing magnetic environment generates electrical energy in the loop, which is at the heart of many electrical devices and technologies.

Biot-Savart Law

The Biot-Savart Law is another fundamental principle in electromagnetism that calculates the magnetic field generated by a current-carrying conductor. Instead of using Ampere's Law, it directly considers the contributions of small segments of the conductor.
The law mathematically depicts how each segment of current generates a magnetic field, expressed as:

• \(dB = \frac{\mu_0}{4\pi} \frac{I \cdot dl \times r}{r^3}\)

where \(I\) is the current, \(dl\) is the current segment's length, \(r\) is the distance to the observation point, and \(\times\) denotes the cross product.
Though not explicitly used in the solution, understanding the Biot-Savart Law provides insight into how magnetic fields can be computed from any wire configurations, helping in complex magnetic field assessments where simpler laws like Ampere's Law become cumbersome.

Lenz's Law

Lenz's Law is an essential concept that complements Faraday's Law. It predicts the direction of the induced current and states that the induced EMF will always oppose the change in magnetic flux that produced it.
The oppositional nature ensures the conservation of energy in electromagnetic systems, and the negative sign in Faraday's equation symbolizes this opposition.
In the context of our problem, Lenz's Law explains why the direction of the induced EMF in the inner loop is negative, as shown in:

• \(V_{induced}(t) = - \pi a^2 \left(\frac{\mu_0 V_0}{2b R_1}\right) \omega \cos \omega t\)

This law ensures that the induced current in the inner loop creates its own magnetic field, opposing the changing magnetic field trying to induce it. Without Lenz's Law, our prediction of the system's behavior would violate basic physical laws, showing its critical role in electromagnetic theory.