Remember the Poynting vector? The magnitude of this vector gives the amount of energy per second that
flows through a unit area (in the direction it gives) because of the presence of a combination of electric and
magnetic field, like in an electro-magnetic wave. The units for the magnitude of the Poynting vector are W/m2
(or J/s/m2
).
Now, it has been shown that this interpretation for the Poynting vector can also apply to any combination of
electric and magnetic fields, even static fields that do not create any electromagnetic wave! This is a somewhat
surprising result, and this homework is about playing with it for one particular example.
Two circular parallel plates are being charged (starting from their uncharged state) with equal but opposite
charges at a constant rate. The two plates have a radius R, are centered around the z axis, and are parallel
to each other, with a distance d between them. Because they are being charged at a constant rate, the electric
field between the two plates increases linearly from 0 until after a time t0 it reaches a final value E0.
1. Calculate the magnitude and direction of the Poynting vector that is found between the plates, at a
distance r from the center, while the plates are being charged.
2. The Poynting vector gives the energy flow per unit time at one particular position. Integrate this energy
flow to find the total energy per unit time that flows into the volume between the plates.
3. What is the total energy that flows into this volume from time 0 to t0 (when the field changes from 0 to
E0)?
4. What is the energy density found in the volume between the circular plates once the electric field has
reached E0?
Quit cheatin boy
1. To calculate the magnitude and direction of the Poynting vector between the plates while they are being charged, we need to consider the electric and magnetic fields in the system.
Since the plates are being charged at a constant rate, the electric field between the plates increases linearly from 0 to E0. At any point between the plates, the electric field is directed from the positive plate towards the negative plate.
However, in this scenario, we have static fields, meaning there are no changing magnetic fields. Therefore, the magnetic field is zero.
The Poynting vector is given by: S = E x B, where E is the electric field and B is the magnetic field.
Since B is zero, the Poynting vector simplifies to: S = E x 0 = 0.
Therefore, the magnitude of the Poynting vector is zero and its direction is undefined.
2. Since the magnitude of the Poynting vector is zero between the plates, there is no energy flow per unit time at any position. Hence, the integration of the energy flow over any volume between the plates will also give zero.
3. Since there is no energy flow during the time period when the field changes from 0 to E0, the total energy that flows into the volume between the plates is zero.
4. Once the electric field has reached E0, the energy density in the volume between the plates can be calculated. Energy density (u) is defined as the energy per unit volume.
In this case, the energy density can be calculated as the total energy divided by the volume between the plates. Since the plates are circular and parallel, we can consider the volume as a cylinder with height d and base area equal to the area of one plate.
The volume V = πR^2d, where R is the radius of the plates and d is the distance between them.
The total energy that flows into the volume between the plates is zero (from the previous answer). Therefore, the energy density will also be zero.
So, the energy density in the volume between the circular plates once the electric field has reached E0 is zero.