Question

The resistance of the loop in the figure is 0.30 Ω.

a) Is the magnetic field strength increasing or decreasing?

b) At what rate (T/s)?

Answer 1

Concepts and reason

The concepts used to solve the problem are Ampere's right-hand rule, Faraday's law of induction, and Ohm's law. First, find the magnetic field strength by using Ampere's right-hand rule. Finally, find the rate of change of magnetic field by comparing the Faradays law and Ohm's law.

Fundamentals

The Ampere's right-hand rule states that "the index finger, middle finger and the thumb are held mutually perpendicular to each other, the index finger gives the direction of the current, middle finger shows the direction of magnetic field and thumb points the direction of force acting on the conducting material". Faraday's law of electromagnetic induction states that, "the induced emf in any closed circuit is equal to the rate of change of magnetic flux linked in that closed circuit". Ohm's law states that, "at a constant temperature, a steady current flows through the conductor is directly proportional to the potential difference developed in the conductor". The expression for Ohm's law is as follows:

\(\varepsilon=I R\)

Here, \(\varepsilon\) is the electromotive force, \(I\) is current, and \(R\) is the resistance. The electromotive force is the rate of change of the magnetic flux. The expression for Faraday's law of induction is as follows:

\(\varepsilon=N \frac{d \Phi}{d t}\)

Here, \(\varepsilon\) is the electromotive force, \(N\) is the number of turns, \(\Phi\) is the magnetic flux, and \(t\) is the time. The area of the square is as follows:

\(A=a^{2}\)

Here, \(a\) is the length and breadth of the square loop, and \(\mathrm{A}\) is the area. The expression for the magnetic flux is as follows:

\(\mathbf{\Phi}=B A\)

Here, \(\mathrm{B}\) is the magnetic field.

(a) The Amperes right-hand rule says that when the right hand of a person curls around the rod, the induced magnetic field due to the current in the loop points out of the page. The external magnetic field is going into the page because of that induced current that creates a magnetic field against the particle's charge. Therefore, the magnetic field strength increases.

The induced current creates a magnetic field that goes against any charge. Due to this, the external magnetic field is increasing. The emf is opposite to that of the magnetic flux linked with the coil.

(b) The expression for Ohm's law is as follows:

\(\varepsilon=I R \ldots \ldots\) (1)

The expression for Faraday's law of induction is as follows:

\(\varepsilon=N \frac{d \Phi}{d t}\)

Replace 1 for \(N\) and \(B A\) for \(\Phi\) in the above equation. \(\varepsilon=-\frac{d(B A)}{d t} \ldots \ldots .\) (2)

Compare equation (1) and (2). \(I R=\frac{d(B A)}{d t}\)

\(I R=A \frac{d B}{d t}\)

\(\frac{d B}{d t}=\frac{I R}{A}\)

Replace \(a^{2}\) for \(\mathrm{A}\) in the above equation. \(\frac{d B}{d t}=\frac{I R}{a^{2}}\)

Substitute \(150 \mathrm{~mA}\) for I, \(0.30 \Omega\) for \(\mathrm{R}\) and \(8.0 \mathrm{~cm}\) for a to find \(d B / d t\). \(\frac{d B}{d t}=\frac{(150 \mathrm{~mA})\left(\frac{10^{-3} \mathrm{~A}}{1 \mathrm{~mA}}\right)(0.30 \Omega)}{(8.0 \mathrm{~cm})^{2}\left(\frac{1 \mathrm{~m}}{100 \mathrm{~cm}}\right)}\)

\(=\frac{\left(150 \times 10^{-3} \mathrm{~A}\right)(0.30 \Omega)}{\left(8.0 \times 10^{-2} \mathrm{~m}\right)^{2}}\)

\(=7.03 \mathrm{~T} / \mathrm{s}\)

The magnetic flux is equal to the product of the magnetic field with the area of the surface. The induced electromotive force is equal to the rate of change of the flux. The induced electromotive force gives a current whose magnetic field opposes the magnetic flux change.

Part a

Thus, the external magnetic field strength will be increased.

Part b

Thus, the rate of change of magnetic field is \(7.03 \mathrm{~T} / \mathrm{s}\).