Question

A metal loop has an area of \(0.100 \mathrm{~m}^{2}\) and is placed flat on the ground. There is a uniform magnetic field pointing due west, as shown in the figure. This magnetic field initially has a magnitude of \(0.123 \mathrm{~T}\), which decreases steadily to \(0.075 \mathrm{~T}\) during a period of \(0.579 \mathrm{~s}\). Find the potential difference induced in the loop during this time.

Short answer

Answer: The potential difference induced in the metal loop is 0.0083 V.

Step-by-step solution

Understand the given information

We have a metal loop with an area of \(0.100 \mathrm{~m}^{2}\). The initial magnetic field inside the loop is \(0.123 \mathrm{~T}\), which decreases to \(0.075 \mathrm{~T}\) over a period of \(0.579 \mathrm{~s}\).

Calculate the initial and final magnetic flux

Magnetic flux (\(\Phi\)) is given by the product of the magnetic field (\(B\)) and the area (\(A\)) of the loop: \(\Phi = BA\). Therefore, we can find the initial and final magnetic flux as: Initial magnetic flux: \(\Phi_i = BA_i = (0.123 \mathrm{~T})(0.100 \mathrm{~m}^2) = 0.0123 \mathrm{~Tm}^2\) Final magnetic flux: \(\Phi_f = BA_f = (0.075 \mathrm{~T})(0.100 \mathrm{~m}^2) = 0.0075 \mathrm{~Tm}^2\)

Calculate the change in magnetic flux

Now let's calculate the change in magnetic flux, which is the final flux minus the initial flux: \(\Delta\Phi = \Phi_f - \Phi_i\) \(\Delta\Phi = 0.0075 \mathrm{~Tm}^2 - 0.0123 \mathrm{~Tm}^2 = -0.0048 \mathrm{~Tm}^2\)

Calculate the rate of change of magnetic flux

To find the rate of change of magnetic flux, we need to divide the change in magnetic flux by the time duration: Rate of change of magnetic flux: \(\frac{d\Phi}{dt} = \frac{\Delta\Phi}{\Delta t} = \frac{-0.0048 \mathrm{~Tm}^2}{0.579 \mathrm{~s}} = -0.0083 \mathrm{~Tm}^2/\mathrm{s}\)

Use Faraday's Law to find the induced potential difference

According to Faraday's Law, the induced potential difference (\(V\)) in the loop is equal to the negative rate of change of the magnetic flux: \(V = -\frac{d\Phi}{dt}\) Therefore, the potential difference induced in the loop is: \(V = -(-0.0083 \mathrm{~Tm}^2/\mathrm{s}) = 0.0083 \mathrm{V}\) (rounded to 2 decimal places) Hence, the potential difference induced in the metal loop during this time is \(0.0083 \mathrm{V}\).

Key concepts

Magnetic Flux

Magnetic flux is a key concept in understanding electromagnetic phenomena. It describes the total magnetic field which passes through a given area. Think of it as the number of magnetic lines of force passing through a surface. For a uniform magnetic field and a flat surface, the magnetic flux (\(Phi\)) can be computed as the product of the magnetic field's strength (\(B\)) and the perpendicular area through which it passes (\(A\)):
\[Phi = BA\]
The unit for measuring magnetic flux is the weber (\(Wb\)), and it forms the foundation for further exploring how changes within this flux can induce a potential difference in nearby conductive materials, as seen in the Faraday's Law of electromagnetic induction.
Just remember, a high magnetic flux means a stronger magnetic field or a larger area is involved, or both!

Induced Potential Difference

When there's a change in magnetic flux through a loop or coil of wire, an electromotive force (EMF) or induced potential difference is generated in that loop. The bigger the change or the quicker it happens, the more voltage is induced. Crucially, this induced potential difference is what makes electric generators work, converting mechanical energy into electrical energy.
According to Faraday's Law, the induced potential difference is directly proportional to the rate of change of the magnetic flux. The mathematical expression for this is quite elegant:
\[V = -\frac{d\Phi}{dt}\]
Where \(V\) is the induced voltage, and \(\frac{d\Phi}{dt}\) is the rate of change of the magnetic flux. The negative sign signifies Lenz's Law, meaning the induced voltage creates a current whose magnetic field opposes the change in the original flux—nature's way of maintaining balance.

Rate of Change of Magnetic Flux

The rate at which the magnetic flux changes, expressed as \(\frac{d\Phi}{dt}\), is a measure of how fast the flux through a loop is varying over time. It's like the speedometer of magnetic flux change! This rate of change is crucial because Faraday’s Law tells us that it's this rate that determines the magnitude of the induced potential difference. In terms of a real-world analogy, think of it like water flowing through a pipe—the faster the water changes speed, the greater the potential energy.
The rate of change can be positive or negative depending on whether the magnetic flux is increasing or decreasing respectively. A greater absolute value of the rate of change equals a greater induced potential difference; hence, fast changes in the magnetic field lead to stronger electrical effects in nearby conductors.