Consider a charging capacitor made out of two identical circular conducting plates of radius 21 cm. The plates are separated by a distance 8 mm (note that d<<a). The bottom plate carries a positive charge


Q(t)= Q0(1+t/T)
with Q0 = 5e-06 C and T = 0.005 sec, and the top plate carries a negative charge -Q(t). The current through the wire is in the positive k-direction. You may neglect all edge effects.

(a) Calculate the components of the electric field (in V/m) inside the capacitor for 2T?


(b) Calculate the components of the magnetic field B (in Teslas) at time 2T inside the capacitor at a distance 8.4 cm from the central axis of the capacitor.


(c) Calculate the components of the Poynting vector S (in W/m²) at time 2T between the plates at a distance r = 8.4 cm from the central axis of the capacitor.

(d) What is the flow of energy (in W) into the capacitor at time 2T ?

(e) How fast is the energy stored in the electric field changing (i.e. what is the rate of change in W) within the capacitor at time 2T ?

its a mystery

Please somebody tell me the answers

(a)

i)0
ii)0
iii)Q(t) = 3 * Q_0
and E = Q(t)/(A * epsilon_0)
....where A = pi*a^2
(b)
i)0
ii)don't know
iii)0
(c)
i)don't know
ii)0
iii)0
(d)
i)don't know
(e)
i)don't know

To calculate the components of the electric field inside the capacitor for 2T, we can use the formula for the electric field between parallel plates:

E = Q / (ε₀A)

where E is the electric field, Q is the charge on the plate, ε₀ is the permittivity of free space, and A is the area of the plate.

Given that Q(t) = Q0(1+t/T), we can substitute this expression into the formula for Q. The plates are identical, so the charge on the bottom plate is Q0(1+t/T) and the charge on the top plate is -Q0(1+t/T).

The electric field between the plates is the sum of the electric fields due to both plates:

E = E(bottom plate) + E(top plate)

The distance between the plates is very small compared to the radius of the plates, so we can assume a uniform electric field between the plates. The field due to the bottom plate is directed upward, and the field due to the top plate is directed downward.

(a) To calculate the components of the electric field inside the capacitor for 2T, we substitute t = 2T into the expression for Q(t) and calculate the electric field using the formula above.

(b) To calculate the components of the magnetic field B at time 2T, we need to consider the current flowing through the wire. The current in the wire generates a magnetic field around it. Since the current is flowing in the positive k-direction, the magnetic field will circulate around the wire in the counterclockwise direction when viewed from above.

Using the right-hand rule (curl your fingers in the direction of the current and your thumb points in the direction of the magnetic field), we can determine the direction of the magnetic field at any point.

At a distance r = 8.4 cm from the central axis of the capacitor, we can calculate the magnetic field using the Biot-Savart law:

B = (μ₀I / 2πr) * sin(θ)

where B is the magnetic field, μ₀ is the permeability of free space, I is the current, r is the distance from the wire, and θ is the angle between the current and the position vector.

(c) To calculate the components of the Poynting vector S at time 2T, we need to calculate the electric field and the magnetic field at that time using the formulas explained in parts (a) and (b). The Poynting vector is given by:

S = E x B

where S is the Poynting vector and x represents the vector cross product.

(d) To calculate the flow of energy into the capacitor at time 2T, we can use the Poynting vector calculated in part (c) to calculate the total power flow through an area A:

P = ∫S · dA

where P is the power, S is the Poynting vector, and the integral is taken over a closed surface enclosing the capacitor plates.

(e) To calculate the rate of change of energy stored in the electric field within the capacitor at time 2T, we need to calculate the total energy stored in the electric field at that time and then differentiate it with respect to time:

dW/dt = d/dt (1/2 ε₀ ∫E² dV)

where dW/dt is the rate of change of energy stored in the electric field, E is the electric field, ε₀ is the permittivity of free space, and the integral is taken over the volume inside the capacitor plates.