Two metal spheres are separated by a distance of 1.1 cm and a power supply maintains a constant potential difference of 710 V between them. The spheres are brought closer to each other until a spark flies between them. If the dielectric strength of dry air is 3.0 106 V/m, what is the distance between the spheres at this time?
To find the distance between the spheres at the time the spark flies between them, we need to consider the breakdown voltage of the air gap between the spheres. The dielectric strength of dry air gives us the maximum electric field that air can withstand before it breaks down and allows a spark to form.
First, let's calculate the breakdown voltage of the air gap:
The breakdown voltage (V_bd) can be calculated using the formula:
V_bd = E * d,
where E is the electric field and d is the distance of the air gap.
Given:
Dielectric strength of dry air (E) = 3.0 x 10^6 V/m
Distance between the spheres (d) = 1.1 cm = 0.011 m
Substituting the values into the formula, we get:
V_bd = (3.0 x 10^6 V/m) * (0.011 m)
V_bd = 33,000 V
Now, since the power supply maintains a constant potential difference of 710 V between the spheres, we need to determine the critical voltage required to bridge the gap and cause a spark:
V_crit = V_bd + 710 V
Substituting the values, we get:
V_crit = 33,000 V + 710 V
V_crit = 33,710 V
Finally, we can find the distance (d_crit) between the spheres at the time the spark flies by rearranging the formula:
d_crit = V_crit / E
Substituting the values, we have:
d_crit = 33,710 V / (3.0 x 10^6 V/m)
d_crit ≈ 0.0112 m
Therefore, the distance between the spheres at the time the spark flies is approximately 0.0112 m.