According to the legend, Zeus, the king of ancient Greek gods, had the power to create thunderbolts in his hands and throw them at his opponents. Suppose Zeus creates the thunderbolts by bringing his palms very close together and parallel to each other and then inducing equal but opposite charge at each of them. What magnitude charge does he have to induce in nC on each palm for this to be possible?
Details and assumptions
Zeus' hands are in air, which has approximately the same dielectric permittivity as vacuum.
Zeus' palms can be modeled as rectangular, with the sides 20 cm and 15 cm.
The threshold for air ionization is E=3×106 V/m
To calculate the magnitude of the charge that Zeus needs to induce on each palm to create thunderbolts, we need to determine the electric field strength required for air ionization.
First, we can calculate the area of Zeus' palm by multiplying the length and width:
Area = 20 cm * 15 cm = 300 cm² = 300 * 10^(-4) m²
Next, we need to calculate the electric field required for air ionization. The electric field is defined as the ratio of the force experienced by a small positive test charge divided by the magnitude of the charge:
E = F / q
Since the force between two charges is given by Coulomb's Law as F = k * (|q1| * |q2|) / r^2, we can rewrite the electric field as:
E = (k * |q2|) / r^2
Where k is the electrostatic constant (k = 8.99 * 10^9 N m²/C²), |q2| is the magnitude of the charge induced on each palm, and r is the separation between the palms.
Given that the threshold for air ionization is E = 3 * 10^6 V/m, we can solve for |q2|:
3 * 10^6 = (8.99 * 10^9 * |q2|) / (0.2 m)
Simplifying the equation, we find:
|q2| = (3 * 10^6 * 0.2 m) / (8.99 * 10^9)
|q2| ≈ 6.68 * 10^(-12) C
To convert this charge to nanoCoulombs (nC), we multiply by 10^9:
|q2| ≈ 6.68 * 10^(-12) C * (10^9 nC / 1 C)
|q2| ≈ 6.68 nC
Therefore, Zeus needs to induce a charge of approximately 6.68 nC on each of his palms to create thunderbolts.