electric fields

Physics II

Class Notes, 3/31/99

The electric field at a point is the force per unit charge experienced by a test charge at that point.

In the figure below we indicate a point A at which we will determine the electric feel due to the 5 `microC point charge at the origin.

The electric field is equal to the force per unit of test charge. We therefore postulate a unit charge at A, and find the force exerted on this unit charge, as indicated below.

The electric field will be equal to the force/charge ratio E = F / q_test.
We see that the electric field strength is 68,000 Newtons/Coulomb at an angle of 29.7 degrees.

Video Clip #01

A similar calculation can be carried out at point B (coordinates (.3m, -.4 m)), except that here we use the fact that that E = F / q_test = k q₁ / r^2 (r / r), so E = k q₁ / r^2.

The distance of B from the origin is `sqrt(.25) m = .5 m.
We obtain an electric field whose strength it 180,000 N / C.
The direction of the force is easily found to be -53 deg.

Video Clip #02

We can visualize the electric field surrounding a positive point charge as in the figure below.

On concentric spheres the distance from the charge is constant so that the strength of the electric field remains constant, with the field directed radially away from the charge.
At the .5 meter distance of point B, the strength in the field is 180 kN / C; this is the field strength over the entire .5 meter sphere centered at the charge.
At the `sqrt(.65) m = .81 m distance, and over the corresponding sphere, the strength of the field is 68 kN / C.

The blue lines radiating from the point charge represent field lines--lines in the direction of the electric field.

The total flux of the charge is 4 `pi k q, analogous to the gravitational flux 4 `pi G m of a mass m. The electric field at a sphere is the flux density over the sphere, or the flux per unit area.
The field lines can be thought of as representing the flux radiating out from the point charge.

The closer we are to the point charge, the denser, or closer together, the field lines are.

The closer the field lines, the more flux there is per unit area.

Thus the closer the field lines are together, the stronger the electric field.

Video Clip #03

The figure below shows the labeling of distances and field strengths more clearly.

In the figure below we have charges q₁ = 5 `microC at (0,0) and (-2 `microC) at (1m ,0). We estimate the electric field at the points A, (.5m, -.5m) and B, (1.2m, -.5m).

The point A is equally distant from the two charges.

At this point the magnitude of electric field E₁ due to the charge q₁ will be greater than that of the electric field E₂ due to the charge q₂, since q₂ has the lesser magnitude. At this point the magnitude of E₂ will be 2/5 that of E₁.

The direction of E₁ will be away from q₁, since q₁ is a positive charge. The direction of E₂ will be toward q₂, since q₂ is negative. (Recall that the direction of electric field is the direction of the force that would be experienced by a positive test charge).

The resultant vector at point A is indicated in red.

Point B is much closer to q₂ than to q₁.

As a result of proximity the electric field E₂ toward q₂ will have a greater magnitude then the field from the more distant charge q₁, even though q₁ is greater in magnitude than q₂ (recall that the field strength due to a charge is inversely proportional to the distance from the charge, and proportional to only the first power of the charge).

Again, E₁ will be directed away from q₁ and E₂ will be toward q₂.

Because of the greater distance to the larger charge q₁, E₁ will have a much smaller magnitude than at point A; E₂ will be somewhat greater than at A, but the resultant vector will still be less than at A.

The resultant vector at point B is indicated in red.

The points A and B lie along the line y = -.5. In the figure below the two resultant vectors are indicated at the appropriate positions on this line.

Several more points are indicated in red on the line.
Before going further you should attempt to show the relative magnitudes and directions of the electric field at each of these points.

Video Clip #04

We see in the figure below that closer to q₁ the magnitude of the resultant gets significantly larger, while the effect of q2 becomes significantly less.

The field E_f is shown as it deviates slightly from the dotted line, which represents the field due to just q1.
The field due to q2 at this point is small, and causes just a slight deviation from the field due to q1.
As we move further from q2, the differenced between the resultant field and the field due to q1 decreas get smaller and smaller..

We see in the figure below how the electric field falls off in the region between q₁ and q₂, becoming larger again near q₂ (though not as large as near q₁) before falling off significantly beyond q₂.

Video Clip #05

In the figure below we have sketched E₁ at several points on a circle about q₁, and E₂ at several points of a circle about q₂.

Before going further, you should at each of the points on the first circle sketch E₂, and at each of the points in the second circle you should sketch E₁. You should then sketch the resultant vector and each point.

In the figure below we have indicated approximate vectors for both fields, and the resultant vector, at two selected points.

Video Clip #06