Question

A perfectly conducting filament is formed into a circular ring of radius \(a\). At one point, a resistance \(R\) is inserted into the circuit, and at another a battery of voltage \(V_{0}\) is inserted. Assume that the loop current itself produces negligible magnetic field. ( \(a\) ) Apply Faraday's law, Eq. (4), evaluating each side of the equation carefully and independently to show the equality; \((b)\) repeat part \(a\), assuming the battery is removed, the ring is closed again, and a linearly increasing \(\mathbf{B}\) field is applied in a direction normal to the loop surface.

Short answer

Answer: With the battery, there is no induced EMF in the loop. Without the battery and with the increasing magnetic field, the induced EMF in the loop is -πa²k, where k is a constant representing the rate of increase of the magnetic field.

Step-by-step solution

Write down Faraday's Law

Faraday's Law states that the induced EMF (\(ε\)) in a loop is equal to the negative time rate of change of the magnetic flux (\(Φ\)) through the loop. Mathematically, this is: \(ε=-\dfrac{dΦ}{dt}\)

Analyze the circuit with the battery

In this scenario, the battery with voltage \(V_0\) is inserted. According to Kirchhoff's Voltage Law, the sum of the potential differences (voltages) around the loop is equal to zero. Considering the battery voltage and the voltage drop across the resistor (Ohm's Law), we can write: \(V_0 - IR - ε = 0\) where \(I\) is the current flowing through the resistor R.

Compute the EMF with the battery

Now, we need to find the induced EMF (\(ε\)) using Faraday's Law. Since the magnetic field is negligible, the rate of change of the magnetic field will be zero. According to Faraday's Law: \(ε = -\dfrac{dΦ}{dt} = 0\) Therefore, there is no induced EMF in the loop with the battery. This implies that: \(V_0 - IR = 0\) Without the battery:

Write the equation for the increasing magnetic field

In this case, it is given that a linearly increasing magnetic field (\(B(t)\)) is applied in a direction normal to the loop surface. We will assume the magnetic field to be in the form: \(B(t) = kt\) where \(k\) is a constant.

Calculate the magnetic flux

Now, we need to find the magnetic flux through the loop due to this magnetic field. The magnetic flux is given by: \(Φ = BAcos(θ)\) Since the magnetic field is perpendicular to the loop, we have \(θ = 0\), and the cosine term will become 1. The area of the circular loop is \(A = πa^2\). Therefore, the magnetic flux becomes: \(Φ(t) = (kt)(πa^2)\)

Calculate the induced EMF from the changing magnetic field

Applying Faraday's Law, we have: \(ε = -\dfrac{dΦ}{dt}\) To find the time derivative of the magnetic flux, we have: \(\dfrac{dΦ(t)}{dt} = πa^2\dfrac{d(kt)}{dt} = πa^2k\) Substituting this into Faraday's Law: \(ε = -\dfrac{dΦ}{dt} = -πa^2k\) This is the induced EMF in the loop without the battery and with the increasing magnetic field.

Key concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental concept in electromagnetism, discovered by Michael Faraday. It refers to the process of generating an electromotive force (EMF) in a conductor due to a changing magnetic field. This principle is captured by Faraday’s Law, which states that the induced EMF in any closed circuit is equal to the negative rate of change of magnetic flux through the circuit.

Imagine a circular loop. When a magnetic field around this loop changes, it can induce an electric current. This phenomena occur in many everyday applications, such as electric generators and transformers.

• Faraday’s Law formula: \( \varepsilon = - \frac{d\Phi}{dt} \)
• The "minus" sign indicates the direction of induced EMF, obeying Lenz's law, stating that the induced current will oppose the change in magnetic flux.

Understanding electromagnetic induction is crucial in fields ranging from physics to engineering, paving the way for the operation of AC electrical systems and technologies that harness the power of changing magnetic fields.

Kirchhoff's Voltage Law

Kirchhoff’s Voltage Law (KVL) plays a pivotal role in understanding how electrical circuits operate. According to KVL, in any closed loop within a circuit, the sum of all the electrical potential differences (voltages) must equal zero.

This principle is beneficial when dealing with complex circuits since it helps in determining unknown voltages. In the exercise, with a battery delivering a voltage of \( V_0 \), and a resistor present, the potential differences across the loop are considered:

• The voltage from the battery: \( V_0 \)
• Voltage drop across the resistor: \( IR \)
• Induced EMF: \( \varepsilon \)

By applying KVL, we can write the equation: \( V_0 - IR - \varepsilon = 0 \), thereby helping us analyze the circuit behavior.
Kirchhoff’s Laws not only simplify the analysis of circuits by using algebra but are fundamental in designing efficient electrical networks and systems.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, considering the strength and the extent of a magnetic field. Think of it as a conceptual tool that quantifies how much magnetic field passes through a specific area.

In practical terms, it is the product of the magnetic field \( B \), the area it penetrates \( A \), and the cosine of the angle \( \theta \) between the field direction and the normal to the surface. This is expressed as:

• Magnetic flux: \( \Phi = B \cdot A \cdot \cos(\theta) \)

In the given exercise, for a circular loop with radius \( a \) and a magnetic field of \( B(t) = kt \) applied normally, the flux \( \Phi(t) \) is calculated based on the area \( \pi a^2 \). Thus, the magnetic flux over time becomes \( \Phi(t) = (kt)(\pi a^2) \).
Grasping the concept of magnetic flux helps in visualizing how magnetic fields interact with physical objects and are crucial for explaining phenomena like electromagnetic induction.

Ohm's Law

Ohm’s Law is a simple yet powerful tool in the study of electrical circuits. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, with the proportionality constant being the resistance of the conductor.

The formula for Ohm's Law is:

• Ohm's Law: \( V = IR \)

Where \( V \) is voltage, \( I \) is current, and \( R \) is resistance. In the exercise, this principle aids in understanding the voltage drop across the resistor when the circuit includes a battery.
When applied in combination with other principles, such as KVL, Ohm's Law becomes indispensable for calculating unknown quantities in electrical circuits and designing reliable electronic and electrical systems. Whether you are setting up a simple electrical circuit or dealing with complex networks, mastering Ohm's Law is key to solving circuit problems effectively.