# Electric field due to distributed planar charge

• easybakeoven
In summary, the problem involves deriving an expression for the electric field E at some distance Y above an annulus with inner and outer radii R1 and R2 and uniform surface charge density σ, lying in the xz plane with the y-axis centered in the hole. The equation E=kQ/r^2 is used as a starting point. The solution involves looking at a small linear slice of the annulus and rotating it around the y-axis, using the cylindrical coordinate system and integrating the expression for the electric field due to a ring of radius dr from r1 to r2.
easybakeoven
An annulus (disk with a concentric hole) has inner and outer radii of R1 and R2 respectively, and uniform surface charge density of $$\sigma$$. the annulus lies in the xz plane with the y-axis centered in the hole.

a.) using the most basic expression of point charge electric field as a starting point, derive an expression for the electric field E at some distance Y above the annulus along the +Y-axis.

E=kQ/r^2

## The Attempt at a Solution

I started working on the problem, and I think that the equation I've come down to for the electric field in the y direction may be close to right...

E_y=K$$\lambda$$2$$\pi$$y*[-1/(r^2+y^2)^(1/2) + 1/(r^2+y^2)^(1/2)]

Is this close to right? I tried to work the problem by looking at a small linear slice of the annulus, and then rotating that around the y-axis

I guess you could do that though I am not sure of the math involved (I think you would use the cylindrical coordinate system to make it easier), Id find the electric field due to a ring of radius dr and then integrate that expression from r1 to r2.

## 1. What is meant by "distributed planar charge" in relation to electric fields?

When we talk about a distributed planar charge, we are referring to a charge that is spread out over a two-dimensional surface. This is in contrast to a point charge, which is concentrated at a single location. In the context of electric fields, the term "distributed" indicates that the charge is not contained at a single point, but rather is spread out over an area.

## 2. How is the electric field due to a distributed planar charge calculated?

The electric field due to a distributed planar charge can be calculated using the principle of superposition. This means that we can break the surface into smaller, easier-to-calculate segments and add up the contributions from each segment to find the total electric field. The formula for the electric field due to a small segment of charge is the same as the formula for the electric field due to a point charge, but we must take into account the distance between the segment and the point where we are measuring the field.

## 3. What is the difference between a uniform and non-uniform distributed planar charge?

A uniform distributed planar charge is one in which the charge is spread out evenly over the entire surface. This means that the charge per unit area is the same everywhere on the surface. In contrast, a non-uniform distributed planar charge has a varying charge density across the surface, meaning that the charge per unit area is not constant. This can make calculating the electric field more challenging, as we must take into account the varying charge density.

## 4. How does the distance from the surface affect the strength of the electric field due to a distributed planar charge?

The electric field due to a distributed planar charge follows an inverse-square law, meaning that as the distance from the surface increases, the strength of the electric field decreases. This is similar to the behavior of the electric field due to a point charge. However, for a distributed planar charge, the field strength also depends on the distance from the surface, with the field being strongest close to the surface and decreasing as we move farther away from it.

## 5. Can a distributed planar charge have a net electric field of zero?

Yes, it is possible for a distributed planar charge to have a net electric field of zero. This can occur when the charge is distributed symmetrically, such that the contributions from each segment of charge cancel each other out. In this case, the electric field at any point above or below the surface would be zero, and the surface would act as an equipotential surface. An example of this is a charged conducting plate with equal and opposite charges on each side.

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