Question

a) Find \(\mathcal{E}(R, t)\), the electromotive force (EMF) around a loop that is at distance \(R\) from the \(z\) axis, where \(R\) is restricted to the region outside the iron rod as shown. Take the direction shown in the figure as positive.

Express \(\mathcal{E}(R, t)\) in terms of \(A ,B_{0}, B_{1}, R,{\text {,and any needed constants such as }} \epsilon_{0}, \pi\), and \(\mu_{0}\)

b)

Due to the cylindrical symmetry of this problem, the induced electric field \(\vec{E}(R, t)\) can depend only on the distance \({n}\) from the \(z\) axis, where \(R\) is restricted to the region outside the iron rod. Find this field.

Express \(\vec{E}(R, t)\) in terms of quantities given in the introduction(and constants), using the unit vectors in the cylindrical coordinate system,\(\hat{\theta} ,\hat{r},\) and \(\hat{z}\)

Answer 1

Concepts and reason

The concept used to solve this problem is magnetic flux and electromotive force. Initially, the electromotive force or induced emf around the loop can be calculated by determining the rate of change of magnetic flux. Later the induced electric field can be calculated by using Faraday's law in one dimension.

Fundamentals

The expression for the electromotive force as a function of time is, \(\varepsilon(t)=-\frac{d \phi_{B}}{d t}\)

Here, \(\varepsilon(t)\) is the induced emf as a function of time, \(d \phi_{B}\) is the change of magnetic flux and \(d t\) is the change in time. The expression for the magnetic flux is, \(\phi_{B}=B A\)

Here, \(\mathrm{B}\) is the magnetic field and \(\mathrm{A}\) is the area of cross-section of the loop of the coil. According to faraday's law, the expression for the electric field is, \(\int E \cdot d l=\varepsilon\)

Here, \(E\) is the induced electric field and \(d l\) is the elementary length.

(a) The variation of the uniform magnetic field inside the rod can be written as, \(B_{Z}(t)=B_{0}+B_{1} t\)

The expression for the electromotive force is,

$$ \varepsilon=-\frac{d \phi}{d t} $$

Substitute \(B A\) for \(\phi\) to find \(\varepsilon\)

$$ \begin{aligned} \varepsilon(t) &=-\frac{d(B A)}{d t} \\ &=-A \frac{d(B)}{d t} \\ \text { Substitute } B_{0}+B_{1} t \end{aligned} $$

The electromotive force or the induced emf around the loop is directly proportional to the rate of change of flux. The flux is the product of the magnetic field and the area of the loop. According to Lenz's law, the induced emf and change in magnetic flux have opposite signs.

(b) The expression for the induced electric field is,

\(\int E \cdot d l=\varepsilon\)

Substitute \(-B_{1} A\) for \(\varepsilon\) and \(2 \pi R\) for \(d l\). \(\int E \cdot d l=-B_{1} A\)

\(E \int d l=-B_{1} A\)

Substitute \(2 \pi R\) for \(\int d l\). \(E(2 \pi R)=-B_{1} A\)

Rearrange the above expression for \(E\). \(E=\frac{-B_{1} A}{2 \pi R} \hat{\theta}\)

The integral of the length element is equal to the circumference of the circle.

The unit vector \(\hat{\theta}\) implies the cylindrical coordinate and the direction of the electric field. The induced electric field depends on the magnetic field, area of the loop, and the distance along the z-axis.

Part a

The electromotive force around the loop is \(-A B_{1}\).

Part \(b\)

The induced electric field is. \(\left(-B_{1} A / 2 \pi R\right) \hat{\boldsymbol{\theta}}\).