Question

A circular loop of area \(A\) is placed perpendicular to a time-varying magnetic field of \(B(t)=B_{0}+a t+b t^{2},\) where \(B_{0}, a,\) and \(b\) are constants. a) What is the magnetic flux through the loop at \(t=0 ?\) b) Derive an equation for the induced potential difference in the loop as a function of time. c) What is the magnitude and the direction of the induced current if the resistance of the loop is \(R ?\)

Short answer

In conclusion: a) The magnetic flux through the loop at t=0 is Φ(0) = B_{0}A. b) The equation for the induced potential difference in the loop as a function of time is EMF(t) = -A(a + 2bt). c) The magnitude and direction of the induced current as a function of time are given by I(t) = (-A(a + 2bt))/R, with the direction of the induced current depending on the change in the magnetic flux (opposing the change if positive and in the direction of the decreasing magnetic field if negative).

Step-by-step solution

Calculate the magnetic flux at t=0

First, we need to find the magnetic field at t=0 by substituting t=0 into the given expression for B(t): B(0) = B_{0} + a(0) + b(0^{2}) = B_{0} Now, the magnetic flux Φ through the loop at t=0 can be calculated using the formula Φ = B⋅A⋅cosθ. Since the loop is perpendicular to the magnetic field, θ = 0°, and cos0° = 1. Φ(0) = B(0)⋅A⋅1 = B_{0}A

Derive an equation for the induced potential difference as a function of time

To find the equation for the induced potential difference (also called electromotive force or EMF) as a function of time, we can use Faraday's law of electromagnetic induction, which states: EMF = -dΦ/dt First, we need to find the magnetic flux Φ as a function of time. Since θ = 0° at all times, we have: Φ(t) = B(t)⋅A = (B_{0}+at+bt^{2})A Now, we can differentiate Φ(t) with respect to time: dΦ/dt = A(dB(t)/dt) = A(a + 2bt) According to Faraday's law: EMF(t) = -dΦ/dt = -A(a + 2bt)

Calculate the magnitude and direction of the induced current

To find the magnitude and direction of the induced current I, we can use Ohm's law: I = EMF/R So, the magnitude of the induced current as a function of time is: I(t) = EMF(t)/R = (-A(a + 2bt))/R The direction of the induced current depends on the direction of the EMF and is determined by Lenz's law. Lenz's law states that the direction of the induced current will always be such that it opposes the change in the magnetic flux that caused it. So, we can observe: - If dΦ/dt > 0, the induced current will be in the opposite direction of the increasing magnetic field. - If dΦ/dt < 0, the induced current will be in the direction of the decreasing magnetic field. Since the sign of the EMF is negative, if (a + 2bt) > 0, the induced current will be in the direction of the increasing magnetic field and if (a + 2bt) < 0, the induced current will be in the opposite direction.

Key concepts

Magnetic Flux

Imagine you're holding a hula hoop in a sea of invisible magnetic field lines. The number of these lines passing through the hoop is what physicists call 'magnetic flux'. Now, let's relate this concept to our circular loop problem. At time zero, because our hoop is perfectly aligned (perpendicular) with the magnetic field, we can easily count how many field lines pass through—this is essentially the magnetic flux. We use the formula \( \Phi = B \cdot A \cdot \cos\theta \) where \( \Phi \) is the magnetic flux, \( B \) is the magnetic field strength, \( A \) is the area of the loop, and \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the loop. Because our loop is oriented perpendicularly to the field, \( \theta \) is zero, and the cosine term simplifies the equation to just \( \Phi = B \cdot A \) as the cosine of zero is one. Hence, the magnetic flux at \( t=0 \) is simply \( B_{0}A \.\)

Faraday's Law

Now, let's dive into Faraday's law of electromagnetic induction, dynamically linking electricity with magnetism. In simplistic terms, Faraday's law tells us that a changing magnetic field within our loop creates an induced voltage, also called electromotive force (EMF). The formula \( EMF = -\frac{d\Phi}{dt} \) indicates this relationship, with the negative sign showing the EMF's direction relative to the flux change, a concept known as Lenz's law (which we'll get to next!). In our circular loop scenario, the time-varying magnetic field changes the flux, and hence, Faraday's law helps predict the EMF induced over time. By differentiating the magnetic flux with respect to time, we can determine how the potential difference in the loop evolves.

Lenz's Law

Lenz's law acts a bit like a superhero's moral code—it won't allow changes without a fight. This physical principle states that any induced current from a change in magnetic flux will create its magnetic field, which fights the original change. It's like a magnetic 'tit for tat.' This protective nature is why the negative sign in Faraday's law is so crucial; it signifies that the induced EMF and, hence, the induced current will always act to oppose the change in magnetic flux. In practical terms, when we see the magnetic flux increasing in our loop, Lenz's law tells us the induced current will flow in a direction to reduce that increase, and vice versa.

Ohm's Law

Last but certainly not least, Ohm's law is like the bread and butter of electric circuits. It says that the current through a conductor between two points is directly proportional to the voltage across the two points. The famous formula \( I = \frac{V}{R} \) describes this relationship, where \( I \) is the current, \( V \) is the voltage (or EMF in the case of induction), and \( R \) is the resistance of the conductor. Applying Ohm's law to our loop with a time-varying magnetic field allows us to calculate the magnitude of the induced current at any time. It's a straightforward yet powerful tool that tells us how much current will flow through our loop, given the induced EMF and loop's resistance.