Question

A shark is able to detect the presence of electric fields as small as $1.0 \mu \mathrm{V} / \mathrm{m} .$ To get an idea of the magnitude of this field, suppose you have a parallel plate capacitor connected to a 1.5 - \(V\) battery. How far apart must the parallel plates be to have an electric field of $1.0 \mu \mathrm{V} / \mathrm{m}$ between the plates?

Short answer

Answer: The distance between the parallel plates must be 1.5 x 10^6 meters.

Step-by-step solution

Write down the formula for electric field in a parallel plate capacitor

The electric field in a parallel plate capacitor can be calculated using the following formula: \(E = \frac{V}{d}\) where \(E\) is the electric field, \(V\) is the voltage, and \(d\) is the distance between the plates.

Plug in the given values

We are given the electric field \(E = 1.0 \mu \mathrm{V} / \mathrm{m}\) and the voltage \(V = 1.5 \, \mathrm{V}\). We can plug these values into the formula: \(1.0 \times 10^{-6} \, \mathrm{V/m} = \frac{1.5 \, \mathrm{V}}{d}\)

Solve for the distance \(d\)

Now we need to solve for \(d\): \(d = \frac{1.5 \, \mathrm{V}}{1.0 \times 10^{-6} \, \mathrm{V/m}}\)

Calculate the distance

Multiply both the numerator and denominator: \(d = \frac{1.5}{1.0 \times 10^{-6}} \, \mathrm{m}\) \(d = 1.5 \times 10^{6} \, \mathrm{m}\)

Write the final answer

The distance between the parallel plates must be \(1.5\times 10^{6}\) meters to have an electric field of \(1.0 \mu \mathrm{V/m}\).