Charge above and infinite grounded plane

In the first case, there is a negative charge that contributes to the energy, while in the conductor case, there is an induced negative charge that does not contribute to the energy. Therefore, the energy for the conductor case is only half of the energy for the first case. This can also be explained by the fact that in the conductor case, the charge is not actually being separated from its original position, but rather being redistributed within the conductor.
  • #1
aaaa202
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The classic image problem is this: Suppose we have a charge q over an infinite grounded plane. What will be the potential at a given point for z>0. It's not just the potential of the charge q since it will induce a certain charge on the conductor.
By looking at the mirror problem of a charges of opposite sign and displaced the same length relative to z=0 but in different directions, one finds from the uniqueness theorem that this solution also holds for the problem above.
Now: According to my book, even though the fields will the same, then the electrostatic energy of the charge is different in the two problems.
In the problem with the conductor the energy is namely only half the energy of the image problem. I have some trouble understanding this. The electrostatic energy is negative because we have to do work to separate the system in both cases. In the first case a negative charge and positive charge are brought in. In the second case a charge q is brought in while the induced charge runs into sustain a zero potential in the conductor. But why would separation require half the energy?
In the first case you would need to pull away the positive charge away while keeping the negative fixed. In the conductor case, what would then happen?
 
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  • #2
aaaa202 said:
The classic image problem is this: Suppose we have a charge q over an infinite grounded plane. What will be the potential at a given point for z>0. It's not just the potential of the charge q since it will induce a certain charge on the conductor.
By looking at the mirror problem of a charges of opposite sign and displaced the same length relative to z=0 but in different directions, one finds from the uniqueness theorem that this solution also holds for the problem above.
Now: According to my book, even though the fields will the same, then the electrostatic energy of the charge is different in the two problems.
In the problem with the conductor the energy is namely only half the energy of the image problem. I have some trouble understanding this. The electrostatic energy is negative because we have to do work to separate the system in both cases. In the first case a negative charge and positive charge are brought in. In the second case a charge q is brought in while the induced charge runs into sustain a zero potential in the conductor. But why would separation require half the energy?
In the first case you would need to pull away the positive charge away while keeping the negative fixed. In the conductor case, what would then happen?

This easiest way to see this is to look at the expression for electrostatic energy in term of the electric field [tex]W=\frac{\epsilon_0}{2}\int_{\text{all space}} E^2 d^3 x[/tex]

The fields for [itex]z\geq0[/itex] are the same for both cases, but what about for [itex]z<0[/itex]?
 

1. What is the concept behind "charge above and infinite grounded plane"?

The concept of "charge above and infinite grounded plane" is a theoretical scenario where a point charge is placed above an infinitely large and grounded conducting plane. This scenario is often used in electrostatics to study the behavior of electric fields and potential due to the presence of the charge.

2. How does the electric field behave in this scenario?

In this scenario, the electric field lines are perpendicular to the conducting plane and are symmetrically distributed around the point charge. The field strength decreases as the distance from the charge increases, following the inverse square law.

3. What is the potential function in this scenario?

The potential function, also known as the electric potential, is the scalar quantity that describes the potential energy per unit charge at a given point in space. In this scenario, the potential function is given by V = kq/r, where k is the Coulomb's constant, q is the magnitude of the point charge, and r is the distance from the charge.

4. How does the potential function change with distance in this scenario?

The potential function decreases with distance following the inverse relationship with the distance, just like the electric field. This means that as the distance from the charge increases, the potential energy per unit charge decreases.

5. What are the implications of this scenario in real-life applications?

This scenario is often used in theoretical models to study the behavior of electric fields and potential in situations where a point charge is placed near a conducting surface. It can also be applied in the design and analysis of electronic devices, such as capacitors and antennas, to understand and optimize their performance.

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