Question

An elastic circular conducting loop expands at a constant rate over time such that its radius is given by \(r(t)=r_{0}+v t\), where \(r_{0}=0.100 \mathrm{~m}\) and \(v=0.0150 \mathrm{~m} / \mathrm{s} .\) The loop has a constant resistance of \(R=12.0 \Omega\) and is placed in a uniform magnetic field of magnitude \(B_{0}=0.750 \mathrm{~T}\), perpendicular to the plane of the loop, as shown in the figure. Calculate the direction and the magnitude of the induced current, \(i\), at \(t=5.00 \mathrm{~s}\).

Short answer

The magnitude of the induced current at t=5s is approximately 0.00103 A, and the direction of the induced current is clockwise within the loop.

Step-by-step solution

Calculate the area of the loop at t=5s

First, we need to find the radius of the loop at t=5s. We can use the given expression for the radius with respect to time: \(r(t)=r_{0}+vt\) \(r(5)=0.100\,\text{m} + 0.0150\,\text{m/s}(5\,\text{s})\) \(r(5)=0.100\,\text{m} + 0.0750\,\text{m} = 0.175\,\text{m}\) Now, we can calculate the area of the loop at t=5s: \(A=\pi r^2\) \(A=\pi (0.175\,\text{m})^2\approx 0.0962\,\text{m}^2\)

Calculate the magnetic flux through the loop

To calculate the magnetic flux through the loop, we can use the following expression: \(\Phi = B_{0}\cdot A\) \(\Phi = 0.750\,\text{T} \cdot 0.0962\,\text{m}^2 \approx 0.0722\,\text{Wb}\)

Apply Faraday's law of electromagnetic induction

Faraday's law states that the induced electromotive force (EMF) is equal to the negative of the time derivative of the magnetic flux: \(EMF = -\frac{d\Phi}{dt}\) To determine the rate of change of the magnetic flux, we will first need to express the magnetic flux as a function of time: \(\Phi(t) = B_{0} \cdot A(t)\) Now, differentiate the magnetic flux with respect to time: \(\frac{d\Phi(t)}{dt} = B_{0}\frac{dA(t)}{dt}\) To find the rate of change of the area A(t), we need to differentiate the area formula with respect to time: \(A(t) = \pi \cdot r^2(t)\) \(\frac{dA(t)}{dt} = 2\pi\cdot r(t) \cdot \frac{dr(t)}{dt} = 2\pi\cdot r(t)\cdot v\) Now, substituting the given values for r(t) and v: \(\frac{dA(t)}{dt}=2\pi(0.175\,\text{m})(0.0150\,\text{m/s})\approx0.0164\,\text{m}^2/\text{s}\) Now, we can calculate the rate of change of the magnetic flux at t=5s: \(\frac{d\Phi(t)}{dt} = B_{0}\frac{dA(t)}{dt}=0.750\,\text{T}(0.0164\,\text{m}^2/\text{s})\approx0.0123\,\text{Wb/s}\) Finally, we can calculate the induced EMF using Faraday's law: \(EMF = -\frac{d\Phi}{dt} = - 0.0123\,\text{Wb/s}\) The negative sign indicates that the direction of the induced current opposes the change in the magnetic field.

Calculate the induced current using Ohm's law

Now that we have the value of the induced EMF, we can calculate the induced current using Ohm's law: \(i = \frac{EMF}{R}\) \(i = \frac{-0.0123\,\text{Wb/s}}{12.0\,\Omega}\approx -0.00103\,\text{A}\)

Determine the direction of the induced current

According to Lenz's law, the induced current will flow in such a direction as to oppose the change in the magnetic field. In this case, since the magnetic field is increasing as the loop expands, the induced current will flow in a clockwise direction.

Conclusion

The magnitude of the induced current at t=5s is approximately 0.00103 A, and it flows in a clockwise direction within the loop.

Key concepts

Faraday's Law

Faraday's Law is a fundamental principle that describes how a voltage can be induced by changing magnetic environments. The law states that the induced electromotive force (EMF) in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. In equation form, it's presented as
\[ EMF = -\frac{d\text{magnetic flux}}{dt} \]

This law is instrumental in understanding how electric generators work and forms the basis of many electrical technologies. Rapid changes in the magnetic field or the movement of a conductor through a static magnetic field can induce this EMF. A negative sign in the equation, known as Lenz's law, signifies that the induced EMF generates a current that opposes the flux change, which is a core concept in electromagnetic induction.

Lenz's Law

Lenz's Law is all about direction – it specifies that the direction of an induced current in a closed conducting loop will always oppose the change in magnetic flux that produced it. Think of it as nature's way of maintaining the status quo. This law complements Faraday's law and is reflected by the negative sign in Faraday's equation.

For instance, if a loop is placed in a magnetic field and the field increases, Lenz's law tells us the induced current will flow in a direction that creates its own magnetic field to counter the increase. If the magnetic field decreases, the induced current's magnetic field will attempt to increase it. This is beautifully illustrated in scenarios where magnets are dropped through conducting tubes and fall more slowly due to the opposing induced currents.

Magnetic Flux

Magnetic flux refers to the measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. The formal definition describes it as the total magnetic field which passes through a given area. It's calculated by the equation:

\[ \text{Magnetic flux} (\text{represented by } \text{Φ}) = B \times A \times \text{cos}\theta \]

where \( B \) is the magnetic field strength, \( A \) is the area the field lines pass through, and \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to \( A \). Magnetic flux is an essential concept for understanding how the change in flux through a conducting loop relates to the induced EMF and resulting current, as dictated by Faraday's Law.

Ohm's Law

Ohm's Law is a cornerstone of electrical engineering and physics, describing the relationship between voltage, current, and resistance in an electrical circuit. Simply put, this law states that the current through a conductor between two points is directly proportional to the voltage across the two points. It is inversely proportional to the resistance between them. The relationship is summed up with the equation:

\[ I = \frac{V}{R} \]

where \( I \) is the current in amperes, \( V \) is the voltage in volts, and \( R \) is the resistance in ohms. Ohm's Law makes it possible to predict how electric currents will behave in a circuit when subjected to different voltages and resistance levels, thus allowing the design and analysis of electronic components and systems.