Question

Two metal spheres are separated by a distance of \(1.0 \mathrm{cm}\) and a power supply maintains a constant potential difference of \(900 \mathrm{V}\) between them. The spheres are brought closer to one another until a spark flies between them. If the dielectric strength of dry air is $3.0 \times 10^{6} \mathrm{V} / \mathrm{m},$ what is the distance between the spheres at this time?

Short answer

Answer: The distance between the spheres when the spark occurs is 0.3 mm or 3.0 x 10^-4 m.

Step-by-step solution

Find the electric field at the point where the spark occurs.

We can find the electric field at the point where the spark occurs by using the dielectric strength of dry air. The dielectric strength represents the maximum electric field that an insulating material, in this case, dry air, can withstand before breaking down. As a result, we can set the electric field at the point of the spark equal to the dielectric strength: \(E_{spark} = 3.0 \times 10^{6} \mathrm{\thinspace V/m}\)

Write down the formula for the electric field and potential difference.

The relationship between electric field and potential difference can be given as: \(V = E \times d\) Where: \(V\) is the potential difference between the two spheres, \(E\) is the electric field at a point in space, \(d\) is the distance between two points. We are given the potential difference between the spheres, \(V = 900 \mathrm{V}\), and we found the electric field at the point of the spark, \(E_{spark}\). Now we need to find the distance, \(d\), between the spheres at that point.

Solve for the distance between the spheres when the spark occurs.

Now let's substitute the values of the potential difference and the electric field into the formula and solve for the distance: \(900 \mathrm{V} = (3.0 \times 10^{6} \mathrm{V/m}) \times d\) To find the distance, we can divide both sides of the equation by \(3.0 \times 10^{6} \mathrm{V/m}\): \(d = \frac{900 \mathrm{V}}{3.0 \times 10^{6} \mathrm{V/m}}\) \(d = 3.0 \times 10^{-4} \mathrm{\thinspace m}\) So, the distance between the spheres when the spark occurs is \(3.0 \times 10^{-4} \mathrm{\thinspace m}\), or \(0.3 \mathrm{mm}\).