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 loop is approximately 0.00829 V.

Step-by-step solution

Find the change in magnetic field

First, let's find the change in magnetic field (\(\Delta B\)) when the initial magnetic field (\(B_i\)) decreases from \(0.123\,\text{T}\) to the final magnetic field (\(B_f\)) value of \(0.075\,\text{T}\): $$ \Delta B = B_f - B_i $$ Substitute the values of \(B_i\) and \(B_f\) into this equation: $$ \Delta B = 0.075\,\text{T} - 0.123\,\text{T} = -0.048\,\text{T} $$ So, the change in magnetic field is \(-0.048\,\text{T}\).

Calculate the change in magnetic flux

Now, we need to calculate the change in magnetic flux (\(\Delta \Phi\)) associated with this change in magnetic field. The magnetic flux is given by the product of the magnetic field and the area (\(A\)) of the loop, perpendicular to the field: $$ \Delta \Phi = \Delta B \cdot A $$ Substitute the values of \(\Delta B\) and the area of the loop: $$ \Delta \Phi = (-0.048\,\text{T}) \cdot (0.100\,\text{m}^2) = -0.0048\,\text{Wb} $$ So, the change in magnetic flux is \(-0.0048\,\text{Wb}\).

Use Faraday's law to find the induced EMF

According to Faraday's law of electromagnetic induction, the induced EMF (\(\varepsilon\)) in the loop is equal to the rate of change of the magnetic flux through the loop: $$ \varepsilon = -\frac{\Delta \Phi}{\Delta t} $$ where \(\Delta t\) is the time period during which the magnetic field changes. Substitute the values of \(\Delta \Phi\) and \(\Delta t\) into the equation: $$ \varepsilon = -\frac{-0.0048\,\text{Wb}}{0.579\,\text{s}} = 0.00829\,\text{V} $$ So, the induced EMF is approximately \(0.00829\,\text{V}\).

Determine the potential difference

Since the induced EMF in the loop is equal to the potential difference across the loop, we can directly say that the potential difference induced in the loop during this time is approximately \(0.00829\,\text{V}\).