Question

A wire of resistance 10 Ω and length 2.5 m is bent into a circle that is concentric with and encircles a solenoid in which the magnetic flux changes from $\mathbf{5}\mathbf{}\mathit{T}\mathbf{.}{\mathit{m}}^{\mathbf{2}}$to $\mathbf{3}\mathbf{}\mathit{T}\mathbf{.}{\mathit{m}}^{\mathbf{2}}$in 0.1 s. What is the emf in the wire? What is the non-Coulomb electric field in the wire? What is the current in the wire?

Short answer

The value of the emf is 20 V, the value of the field is$8\frac{V}{m}$ and the value of the current in the wire is 2 A.

Step-by-step solution

Write the given data from the question.

The resistance of the wire is$10\Omega$ .

The length of the wire is$2.5m$ .

The change in the magnetic flux is:

.$3\text{\hspace{0.33em}T}\cdot {\text{m}}^2-5\text{\hspace{0.33em}T}\cdot {\text{m}}^2=-2\text{\hspace{0.33em}T}\cdot {\text{m}}^2$

The time taken to change the flux is 0.1 s

Determine the formula

Consider the formula for the change in magnetic flux as:

$emf=-\frac{d\varphi }{dt}$

Derive the formula for the non-columbic electric field as:

$\begin{array}{l}emf=\int E\cdot dl\\ emf=El\cos \theta \\ em{f}_{\max }=El\\ E=\frac{em{f}_{\max }}{l}\end{array}$

Determine the value of emf in the wire.

Substitute the value of emf in the wire and solve as:

$\begin{array}{c}emf=\frac{-2\text{\hspace{0.33em}T}\cdot {\text{m}}^2}{0.1\text{\hspace{0.33em}s}}\\ =20\text{\hspace{0.33em}V}\end{array}$

The emf of the wire is 20V

Determine the value of non-coulomb electric field in the wire.

Substitute the value in the formula for the for the non-columbic field and solve as:

$\begin{array}{c}E=\frac{20\text{\hspace{0.33em}V}}{2.5\text{\hspace{0.33em}m}}\\ =8\frac{\text{V}}{\text{m}}\end{array}$

The value of the electric field is$8\frac{\text{V}}{\text{m}}$

Determine the value of current in the wire

From the ohm’s law solve for the value of the current in the wire as:

$\begin{array}{c}I=\frac{V}{R}\\ =\frac{20\text{\hspace{0.33em}V}}{10\Omega }\\ =2\text{\hspace{0.33em}A}\end{array}$

The value of the current is 2 A