Question

A rectangular conducting loop with dimensions \(a\) and \(b\) and resistance \(R\), is placed in the \(x y\) -plane. A magnetic field of magnitude \(B\) passes through the loop. The magnetic field is in the positive \(z\) -direction and varies in time according to \(B=B_{0}\left(1+c_{1} t^{3}\right),\) where \(c_{1}\) is a constant with units of \(1 / \mathrm{s}^{3}\) What is the direction of the current induced in the loop, and what is its value at \(t=1 \mathrm{~s}\) (in terms of \(a, b, R, B_{0},\) and \(\left.c_{1}\right) ?\)

Short answer

Answer: The direction of the induced current is clockwise, and its value at \(t = 1\,\text{s}\) is \(\frac{3abB_0c_1}{R}\).

Step-by-step solution

Find the change in magnetic flux

The magnetic flux through the loop is given by \(\Phi = \int\int \vec{B} \cdot \mathrm{d} \vec{A}\). Since the magnetic field is uniform and normal to the loop, \(\Phi = B(t) ab\). We need to find the time rate of change of the magnetic flux, which is: $$\frac{d\Phi}{dt} = ab\frac{dB}{dt}$$

Calculate the induced electromotive force (EMF)

According to Faraday's law, the induced EMF in a loop is equal to the negative rate of change of magnetic flux through the loop: $$\text{EMF} = -\frac{d\Phi}{dt} = -ab\frac{dB}{dt}$$

Determine the time derivative of the magnetic field

Differentiating the given expression for \(B\) with respect to time \(t\): $$ \frac{dB}{dt} = B_0(3c_1t^2)$$

Calculate the induced EMF at \(t = 1\,\text{s}\)

Plug in the expression found in Step 3 into the equation found in Step 2, for \(t = 1\,\text{s}\): $$\text{EMF} = -ab(-3B_0c_1) = 3abB_0c_1$$

Find the induced current

According to Ohm's law, the current in a resistor is equal to the voltage divided by the resistance: $$I = \frac{\text{EMF}}{R} = \frac{3abB_0c_1}{R}$$

Determine the direction of the induced current

Since the time derivative of the magnetic field is positive (\((3c_1t^2)\)), the magnetic field is increasing in the positive \(z\)-direction. By Lenz's law, the induced current will generate a magnetic field that opposes the increase in the external magnetic field. Using the right-hand rule, we can find that the induced current will be clockwise when viewed from the positive \(z\)-axis. The direction of the induced current is clockwise, and its value at \(t = 1\,\text{s}\) is \(\frac{3abB_0c_1}{R}\).

Key concepts

Faraday's Law

Imagine you have a coil of wire and a magnet. If you move the magnet through the coil, you'll notice that electricity starts to flow in the wire without any batteries connected. That's because of a phenomenon called electromagnetic induction, and it's explained by Faraday's Law.

Michael Faraday discovered that a changing magnetic field within a closed loop of wire induces an electromotive force (EMF) in the wire. The law states that the induced EMF is equal in magnitude and opposite in sign to the rate of change of the magnetic flux through the loop. This law can be mathematically expressed as: ewlineewline \[ \text{EMF} = -\frac{d\Phi}{dt} \]

• The minus sign is crucial—it's a reflection of Lenz's Law, which tells us the direction of the induced current.
• EMF can be thought of as the 'pressure' that pushes electrons to create current.

Using Faraday's Law, we can calculate the EMF in any circuit with a changing magnetic environment, helping us to understand how generators, transformers, and even wireless charging works.

Lenz's Law

Lenz's Law is best friends with Faraday's Law, but it focuses on direction rather than quantity. It was formulated by Heinrich Lenz and states that the current induced by a change in magnetic field will flow in such a direction that it will oppose the change that produced it.

This law is like the universe's way of applying the brakes. If you have a magnetic field that's increasing in strength, Lenz's Law says, 'Hold on, let's create a current that makes a magnetic field to push against this increase.'

It's beautifully symmetric and keeps energy conservation in check, ensuring that you can't get energy out of nowhere. The direction of the induced current is determined using the right-hand rule: point your thumb in the direction of the magnetic field, and your fingers will curl in the direction of the current. Lenz's Law is what keeps the balance in the world of electromagnetism, preventing the magnetic equivalent of perpetual motion machines.

Magnetic Flux

Magnetic flux is a bit like the amount of water flowing through a pipe. Instead of water, it's the magnetic field, and instead of a pipe, it's an area through which the field lines are passing. Specifically, the magnetic flux through a surface area is the product of the magnetic field and the area of the surface through which the field penetrates, as well as the cosine of the angle between the field lines and the normal (perpendicular) to the surface.

We can write this down as: ewlineewline \[ \Phi = \int\int \vec{B} \cdot \mathrm{d} \vec{A} \]

• The symbol \( \Phi \) (phi) represents the magnetic flux.
• \( \vec{B} \) represents the magnetic field, and \( \mathrm{d} \vec{A} \) is a tiny bit of area with the orientation of the surface.

In simple cases where the magnetic field is uniform and at a right angle to the area, the equation simplifies to \( \Phi = B \cdot A \). This concept is central to understanding electromagnetic induction as it links the magnetic field to the EMF and current produced in a conductor.

Ohm's Law

Ohm's Law is like the golden rule for electric circuits, providing a fundamental principle that governs the flow of current. This law, named after Georg Simon Ohm, tells us how voltage, current, and resistance interact with each other: ewlineewline \[ I = \frac{V}{R} \]

• \(I\) is the current flowing through the conductor in amperes (A).
• \(V\) is the voltage measured across the conductor in volts (V).
• \(R\) is the resistance of the conductor in ohms (\(\Omega\)).

If you know any two of the quantities, you can calculate the third. In the context of electromagnetic induction, Ohm's Law helps us find out the current induced by the EMF. It's like knowing how much traffic will flow down a street if you know the speed limit (voltage) and the number of lanes (resistance). Ohm's Law is an incredibly powerful tool in both designing and analyzing electrical circuits.