What is the maximum charge that can be put on a conducting sphere of radius 15 cm if it is not to spark? That is, the electric field at its surface cannot exceed the dielectric strength of air.
b) What is the sphere's voltage at this charge?
r=0.15 m, E< E₀
The dielectric strength of air E₀ =3 •10⁻³ V/m
E=q/4πε₀r² =>
q=4πε₀r²E = 4πε₀r²E₀=
= 4π•8.85•10⁻¹²•0.15²•3 •10⁻³=7.5•10⁻¹⁵ C
To find the maximum charge that can be put on the conducting sphere without it sparking, we need to consider the electric field at its surface. The electric field at the surface of a conducting sphere given its charge (q) and radius (r) can be calculated using the equation:
E = k * (q / r²)
Where:
E is the electric field,
k is the electrostatic constant (8.99 x 10^9 Nm²/C²),
q is the charge, and
r is the radius of the sphere.
Since the electric field cannot exceed the dielectric strength of air, we need to find the maximum electric field value. The dielectric strength of air is approximately 3 x 10^6 N/C. Setting E equal to the dielectric strength and rearranging the formula, we can solve for the maximum charge:
q = E * r² / k
Substituting the given values into the equation:
q = (3 x 10^6 N/C) * (0.15 m)² / (8.99 x 10^9 Nm²/C²)
Calculating this expression will give us the maximum charge that can be placed on the sphere without sparking.
To find the sphere's voltage at this charge, we can use the equation:
V = k * (q / r)
Where:
V is the voltage.
Substituting the known values into the formula, we can calculate the voltage.
Please note that in scientific calculations, it's important to use consistent units. In this case, we've converted the radius from cm to m to ensure consistent SI units.