Faraday's Law with Calculus: Videos & Practice Problems

Faraday's Law with Calculus connects a changing magnetic flux to an induced EMF at a specific instant in time. The average form becomes the instantaneous form \(|\varepsilon| = N\left|\frac{d\phi_B}{dt}\right|\), where the flux is \(\phi_B = BA\cos\theta\). This means the induced EMF depends on how fast the flux changes, not just on the flux itself.

To apply the calculus form, identify which part of \(\phi_B = BA\cos\theta\) changes with time. If the magnetic field changes, then \(\frac{d\phi_B}{dt}=A\cos\theta\\,\frac{dB}{dt}\). If the area changes, then \(\frac{d\phi_B}{dt}=B\cos\theta\\,\frac{dA}{dt}\). If the angle changes, then \(\frac{d\phi_B}{dt}=BA\\,\frac{d}{dt}(\cos\theta)\). For rotation, the angle is often written as \(\theta=\omega t\), so the derivative is found using the chain rule.

Faraday's Law with Calculus

Faraday's law of electromagnetic induction relates a changing magnetic flux to an induced electromotive force (EMF). When analyzing how the magnetic flux changes over time, calculus provides a powerful tool to determine the instantaneous EMF rather than just an average value. The fundamental equation for Faraday's law is given by:

\[\mathcal{E} = -N \frac{\Delta \Phi}{\Delta t}\]

where 𝔈 is the induced EMF, N is the number of turns in the coil, and Φ is the magnetic flux. As the time interval Δt approaches zero, the average rate of change becomes an instantaneous rate of change, expressed as a derivative:

\[\mathcal{E} = -N \frac{d\Phi}{dt}\]

The magnetic flux Φ through a loop is defined as:

\[\Phi = B A \cos \theta\]

where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal (perpendicular) to the loop's surface.

Depending on which variable changes with time, the derivative of the flux can be expanded accordingly:

• If the magnetic field B changes with time, while A and θ remain constant:

\[\mathcal{E} = -N A \cos \theta \frac{dB}{dt}\]

• If the area A changes with time, while B and θ remain constant:

\[\mathcal{E} = -N B \cos \theta \frac{dA}{dt}\]

• If the angle θ changes with time, while B and A remain constant:

\[\mathcal{E} = N B A \frac{d}{dt}(\cos \theta)\]

These forms allow for calculating the instantaneous induced EMF by identifying the time-dependent variable in the problem.

For example, consider a circular loop of radius 2 meters lying flat, with a magnetic field passing perpendicularly through it. The magnetic field varies with time according to the equation:

\[B(t) = 4 + 3t^2\]

Since the magnetic field B changes with time, and the loop's area and orientation remain constant, the induced EMF is calculated by:

\[\mathcal{E} = -N A \cos \theta \frac{dB}{dt}\]

Given that the loop has a single turn (N = 1), the area of the circular loop is:

\[A = \pi r^2 = \pi \times 2^2 = 4\pi\]

The magnetic field is perpendicular to the loop, so the angle θ = 0° and thus:

\[\cos \theta = \cos 0° = 1\]

Taking the derivative of the magnetic field with respect to time:

\[\frac{dB}{dt} = \frac{d}{dt}(4 + 3t^2) = 0 + 6t = 6t\]

Substituting these values into the EMF equation yields:

\[\mathcal{E} = -1 \times 4\pi \times 1 \times 6t = -24\pi t\]

The negative sign indicates the direction of the induced EMF according to Lenz's law, but the magnitude is:

\[|\mathcal{E}| = 24\pi t \text{ volts}\]

At a specific time, for example t = 0.5 seconds, the induced EMF magnitude is:

\[|\mathcal{E}| = 24\pi \times 0.5 = 12\pi \approx 37.7 \text{ volts}\]

This example illustrates how calculus enables the calculation of instantaneous induced EMF when the magnetic flux changes non-uniformly over time. Understanding the relationship between magnetic flux and induced EMF, and applying derivatives to time-dependent variables, is essential for solving advanced electromagnetic induction problems.

Faraday's Law with Calculus Example 1

When a rectangular conducting loop with length A and width B is placed in a uniform magnetic field and rotated about its axis at an angular velocity ω, an electromotive force (EMF) is induced in the loop. To determine the induced EMF as a function of time, we apply Faraday's law of electromagnetic induction, which states that the induced EMF is equal to the negative rate of change of the magnetic flux through the loop.

The magnetic flux Φ through the loop is given by the product of the magnetic field B, the area of the loop A × B, and the cosine of the angle θ between the magnetic field and the normal to the loop's surface:

\[\Phi = B \cdot A \cdot B \cdot \cos \theta\]

Since the magnetic field B and the loop's dimensions A and B remain constant, the only variable changing with time is the angle θ. As the loop rotates at angular speed ω, the angle varies as:

\[\theta = \omega t\]

Using Faraday's law, the induced EMF ε is:

\[\varepsilon = -N \frac{d\Phi}{dt} = -N B A B \frac{d}{dt} \cos(\omega t)\]

Here, N is the number of loops, which is 1 in this case. Applying the chain rule to differentiate the cosine function:

\[\frac{d}{dt} \cos(\omega t) = -\omega \sin(\omega t)\]

Substituting back, the induced EMF becomes:

\[\varepsilon = -1 \times B \times A \times B \times (-\omega \sin(\omega t)) = \omega A B B \sin(\omega t)\]

To clarify, the area of the loop is A × B, so the expression simplifies to:

\[\varepsilon(t) = \omega B A B \sin(\omega t)\]

This equation shows that the induced EMF varies sinusoidally with time, oscillating as the loop rotates within the magnetic field. The amplitude of the EMF depends on the angular velocity ω, the magnetic field strength B, and the loop's area A × B. This relationship is fundamental in understanding how rotating coils generate alternating current in electric generators.

Here's what students ask on this topic:

What is the calculus form of Faraday's Law and how is it derived?

Faraday's Law in calculus form relates the instantaneous induced electromotive force (EMF) to the rate of change of magnetic flux. The average EMF is given by $\epsilon =N\frac{\mathrm{\Delta \phi }}{\mathrm{\Delta t}}$, where $\epsilon$ is EMF, $N$ is the number of turns, and $\mathrm{\delta \phi }$ is the change in magnetic flux over time $\mathrm{\delta t}$. As $\mathrm{\delta t}$ approaches zero, this becomes the instantaneous rate of change, or derivative:

$\epsilon =N\left|\frac{\mathrm{d\phi }}{\mathrm{dt}}\right|$

Here, magnetic flux $\phi$ is $BA\cos \theta$, where $B$ is magnetic field, $A$ is area, and $\theta$ is the angle between the field and the normal to the loop. This form allows calculation of EMF at any instant by differentiating the flux with respect to time.

How do you calculate the induced EMF when the magnetic field changes with time?

When the magnetic field $B$ changes with time, but the area $A$ and angle $\theta$ remain constant, the induced EMF is found by differentiating the magnetic flux:

$\epsilon =N\left|\frac{\phi }{\mathrm{dt}}\right|=NA\cos \theta \left|\frac{B}{\mathrm{dt}}\right|$

Here, $N$ is the number of turns, and the derivative $\frac{B}{\mathrm{dt}}$ represents how fast the magnetic field changes with time. This formula shows that the induced EMF depends directly on the rate of change of the magnetic field, scaled by the loop's area and orientation.

What is the method to find induced EMF when the loop area changes over time?

If the loop's area $A$ changes with time while the magnetic field $B$ and angle $\theta$ remain constant, the induced EMF is calculated by differentiating the flux with respect to time as:

$\epsilon =NB\cos \theta \left|\frac{A}{\mathrm{dt}}\right|$

This means the EMF depends on how quickly the area changes. For example, if the loop is stretched or compressed, changing its area, this formula helps find the instantaneous EMF induced due to that change.

How do you calculate induced EMF when the angle between the magnetic field and the loop changes?

When the angle $\theta$ between the magnetic field and the loop's normal changes with time, while the magnetic field $B$ and area $A$ remain constant, the induced EMF is given by:

$\epsilon =NBA\left|\frac{d\mathrm{dt}}{}\cos \right(\theta \left)\right|$

Using the chain rule, if $\theta =\omega t$ (rotating loop with angular velocity $\omega$), then:

$\frac{d\mathrm{dt}}{}\cos \left(\theta \right)=\; -\omega \sin \left(\theta \right)$

This shows the EMF varies sinusoidally with time due to rotation.

Can you provide an example of calculating instantaneous EMF using Faraday's Law with calculus?

Consider a circular loop of radius 2 m in a magnetic field $B=\; 4\; +\; 3t^2$ (in teslas), where $t$ is time in seconds. The loop lies flat with the magnetic field perpendicular to it, so $\theta =\; 0$ and $\cos \left(0\right)=\; 1$. The number of turns $N=\; 1$.

The area is $A=\; C0\; r^2\; =\; C0\; imes\; 2^2\; =\; 4C0$.

The induced EMF is:

$\epsilon =\; N\; A\; C6\text{'}\; =\; 1\; imes\; 4C0\; imes\; rac\{dB\}\{dt\}$

Calculate $\frac{\mathrm{dB}}{\mathrm{dt}}$:

$\frac{d}{\mathrm{dt}}(4\; +\; 3t^2)\; =\; 6t$

At $t=\; 0.5$ s,

$\epsilon =\; 4C0\; imes\; 6\; imes\; 0.5\; =\; 12C0\; ext\{\; volts\}$

This is the instantaneous EMF at 0.5 seconds.