The capacitance of a spherical capacitor with inner radius b and outer radius a.
Suppose a is fixed. What is radius b in terms of a such that the capacitor can store a maximum amount of energy , assuming that the electric field strength at the surface of the inner sphere shall not greater than E0 ? And what is this maximum energy?
To find the radius b in terms of a such that the capacitor can store a maximum amount of energy while ensuring that the electric field strength at the surface of the inner sphere is not greater than E0, we need to consider the relationship between capacitance (C), energy (U), and electric field intensity (E) for a spherical capacitor.
The capacitance of a spherical capacitor can be calculated using the formula:
C = 4πε₀(ab) / (a - b)
Where ε₀ is the permittivity of free space.
The energy stored in a capacitor can be calculated using the formula:
U = (1/2)CV²
Where V is the potential difference across the capacitor.
Now, to maximize the energy stored in the capacitor while ensuring that E ≤ E₀, we minimize the capacitance. In order to minimize the capacitance, we want to maximize the denominator (a - b) in the formula.
Since a is fixed, to maximize (a - b), we need to minimize b as much as possible. This means that b approaches zero.
So, the radius b in terms of a to maximize the energy stored is:
b = 0
This means the inner sphere radius is reduced to zero.
Now, to calculate the maximum energy (U), we substitute b = 0 into the formula for capacitance:
C = 4πε₀(a(0)) / (a - 0) = 0
Since capacitance is zero, the maximum energy stored would also be zero.
Therefore, the maximum energy stored is zero when the inner sphere radius is reduced to zero.
To find the radius b in terms of a such that the capacitor can store a maximum amount of energy, we need to consider the relationship between capacitance and energy.
The capacitance C of a spherical capacitor is given by the formula:
C = 4πε₀ab / (a - b)
where ε₀ is the permittivity of free space, a is the outer radius, and b is the inner radius.
To find the maximum energy (U) that can be stored in a capacitor, we use the formula:
U = (1/2)CV²
where V is the potential difference across the capacitor.
Now, let's consider the condition that the electric field strength at the surface of the inner sphere shall not be greater than E₀. The electric field E inside a spherical capacitor is given by:
E = Q / (4πε₀r²)
where Q is the charge on the inner sphere and r is the distance from the center of the capacitor.
To ensure that the electric field at the surface of the inner sphere is not greater than E₀, we set E = E₀ and substitute the values for Q and r in terms of a and b:
E₀ = Q / (4πε₀b²)
From this equation, we can solve for Q:
Q = 4πε₀b²E₀
Now, the potential difference V across the capacitor is given by:
V = Q / (4πε₀a)
Substituting the value of Q, we get:
V = (4πε₀b²E₀) / (4πε₀a)
Simplifying, we find:
V = b²E₀ / a
Substituting this value of V into the formula for U, we obtain:
U = (1/2)C(b²E₀ / a)²
Now, to determine the maximum energy, we differentiate U with respect to b and find the critical points by setting the derivative equal to zero:
dU / db = 0
Solving this equation will give us the value of b that maximizes U.