Electric field of a dipole

In summary, we discussed the problem of finding the bound charge and sketching the electric field for a short cylinder with frozen-in uniform polarization. The bound charge in this case is zero, and the electric field varies depending on the length of the cylinder compared to its radius. For L >> a, the electric field falls off like 1/r, for L << a, it falls off like 1/r^3, and for L approximately equal to a, it transitions between these two behaviors resulting in a concave downwards curve.
  • #1
stunner5000pt
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Griffiths problem 4.11 page 170
A short cylinder of radius a and length L carries a frozen in uniform polarization P parallel to its axis. Find the bound charge and sketch theelectric field for
L>> a
L<<a
L approximately equal to a

well i have no problem finding the bound charge but the electricfield.. would in eed to find an explicit expression or just sketch it qualitatively??

Suppoose L << a then it look a dipole doesn't it and the field falls off like 1/r^3
suppose L >> a then it looks like an infinite rod and the electric field falls off like 1/r

but For L approx equal to a I am stumped...

please help!

than kyou for all teh advice!
 
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  • #2


Hello, thank you for your question. I can offer some insight on this problem.

First, let's start with the bound charge. As you mentioned, finding the bound charge is not a problem. The bound charge in this case would be zero, since the cylinder is frozen with a uniform polarization parallel to its axis. This means that there are no free charges on the surface of the cylinder.

Now, for the electric field. To sketch the electric field, we need to consider the electric field at different points around the cylinder. For L >> a, as you mentioned, the cylinder looks like an infinite rod. In this case, the electric field would be radial and would fall off like 1/r, where r is the distance from the axis of the cylinder.

For L << a, the cylinder looks like a dipole, as you mentioned. In this case, the electric field would be radial and would fall off like 1/r^3, where r is the distance from the axis of the cylinder.

For L approximately equal to a, the electric field would be a combination of these two cases. Near the ends of the cylinder (i.e. at points close to the surface), the electric field would behave like that of a dipole, falling off like 1/r^3. But as we move towards the center of the cylinder, the electric field would start to resemble that of an infinite rod, falling off like 1/r. This would result in a smoother transition between the two behaviors.

To sketch this, you could draw a graph of the electric field strength (E) vs. distance from the axis (r). Near the ends of the cylinder, the graph would have a steeper slope (falling off like 1/r^3), and as you move towards the center, the slope would decrease (falling off like 1/r). This would result in a curve that is concave downwards, with the highest point being at the center of the cylinder.

I hope this helps. Good luck with your problem!
 

1. What is an electric dipole?

An electric dipole is a pair of equal and opposite electrical charges that are separated by a small distance. This creates a dipole moment, which is a vector quantity that represents the strength and direction of the dipole.

2. How is the electric field of a dipole calculated?

The electric field of a dipole can be calculated using the equation E = kq/r^2, where k is the Coulomb constant, q is the magnitude of the charge, and r is the distance from the dipole. The direction of the electric field is toward the positive charge and away from the negative charge.

3. What is the significance of the dipole moment in an electric field?

The dipole moment is a measure of the separation of the charges in a dipole and plays a critical role in determining the strength and direction of the electric field. It is also used to calculate the potential energy of a dipole in an electric field.

4. How does the electric field of a dipole change with distance?

The electric field of a dipole decreases as the distance from the dipole increases. This can be seen in the inverse square relationship in the equation E = kq/r^2, where the electric field is inversely proportional to the square of the distance from the dipole.

5. Can an electric dipole exist in a vacuum?

Yes, an electric dipole can exist in a vacuum. In fact, many natural and man-made systems exhibit electric dipoles, such as atoms, molecules, and antennas. The concept of an electric dipole is an important part of understanding the behavior of electric fields in various systems.

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