Question

A circular loop of area \(A\) is placed perpendicular to a time-varying magnetic field of magnitude \(B(t)=B_{0}+a t+b t^{2},\) 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 are the magnitude and the direction of the induced current if the resistance of the loop is \(R ?\)

Short answer

Answer: The magnetic flux at \(t=0\) is \(B_0 \cdot A\).

Step-by-step solution

Determine Magnetic Field at t=0

We are given the magnetic field as a function of time, \(B(t) = B_0 + at + bt^2\). To find the magnetic field at \(t=0\), plug in \(t=0\) into the equation: \(B(0) = B_0 + 0 + 0 = B_0\)

Calculate the Magnetic Flux at t=0

The magnetic flux through the circular loop is given by the formula \(\Phi = B\cdot A\) where \(A\) is the area of the loop and \(B\) is the magnetic field at time \(t=0\). Since the loop is perpendicular to the magnetic field, we have: \(\Phi (0) = B(0) \cdot A = B_0 \cdot A\) So, the magnetic flux at \(t=0\) is \(B_0 \cdot A\). #b) Induced Potential Difference Equation#

Calculate the Time Derivative of Magnetic Field

To calculate the induced electromotive force (EMF) using Faraday's Law of Electromagnetic Induction, we need to find the time derivative of the magnetic field which is given by \(B(t) = B_0 + at + bt^2\). Differentiate \(B(t)\) with respect to time \(t\): \(\frac{dB(t)}{dt} = 0 + a + 2bt\)

Apply Faraday's Law to Find the Induced EMF

Faraday's law of electromagnetic induction states that the induced EMF (\(\varepsilon\)) is equal to the negative rate of change of the magnetic flux: \(\varepsilon = -\frac{d\Phi}{dt}\) Now, substitute \(\Phi = B(t) \cdot A\) into Faraday's law and find its time derivative: \(\varepsilon = -A\frac{dB(t)}{dt} = -A(a + 2bt)\) So, the equation for the induced potential difference (EMF) in the loop as a function of time is \(\varepsilon (t) = -A(a+2bt)\). #c) Induced Current and Direction#

Calculate the Induced Current

With the induced EMF and the resistance of the loop \(R\), we can use Ohm's law to find the induced current: \(I = \frac{\varepsilon}{R}\) Using the equation for the induced EMF from part (b): \(I(t) = \frac{-A(a+2bt)}{R}\) Thus, the magnitude of the induced current as a function of time is given by \(|I(t)| = \frac{A|a+2bt|}{R}\).

Determine the Direction of the Induced Current

To determine the direction of the induced current, we need to consider the sign of the induced EMF and Lenz's law. The negative sign in our equation for the induced EMF, \(\varepsilon (t) = -A(a + 2bt)\), indicates the direction is such that it opposes the change in magnetic flux through the loop. If the change in magnetic field is positive (increasing), the induced current will flow in a direction to create a magnetic field opposing the increase. If the change in magnetic field is negative (decreasing), the induced current will flow in a direction to create a magnetic field opposing the decrease. Therefore, the direction of the induced current depends on the sign of \((a + 2bt)\), which determines whether the magnetic field is increasing or decreasing. If \((a + 2bt) > 0\), the current will flow in one direction, and if \((a + 2bt) < 0\), the current will flow in the opposite direction.