5.4.1: Illustrations

Illustration 1: Energy and Voltage

The animation shows a positive test charge in a uniform electric field created by two parallel plates of constant charge. You can drag the test charge between the plates to any spot and then push "set value and play" to see it move. You can also read position, voltage, and magnitude of the electric field. The graph plots the kinetic energy and potential energy as a function of height above the bottom plate. Restart.

• Does the charge experience a constant force? Explain.
• As the charge moves, is the work done on it by the electric force positive or negative? Explain.
• As you move the charge to a different starting place, how does the total energy change?
• To what positions can you move the charge to decrease the total energy, to increase it, and to keep it the same?

The charge experiences a constant force since the potential energy curve is linear (as with gravity). Since the kinetic energy increases, the work done by the electrostatic force is positive (and the change in potential energy is negative). Since work here is \(\mathbf{F}\cdot\Delta\mathbf{x}\), the larger the displacement, the larger the work done and the larger the change in kinetic energy.

The potential energy divided by the charge of the particle is the electric potential (measured in Volts). Triple the charge of the center particle. In this new configuration the potential energy increases, but the electric potential remains the same (for the charge at the same position). Why?   What would need to happen to change the electric potential (voltage)?

Illustration authored by Anne J. Cox.
Script authored by Wolfgang Christian and Anne J. Cox.

Illustration 2: Work and Equipotential Surfaces

The animation shows the equipotential contours around a charge distribution. Changes in the bar graph height show the work done to move the red test charge (position is given in meters, electric potential is given in volts and work is given in microjoules). Equipotential surfaces are simply surfaces (in this two-dimensional representation they are lines) of constant electric potential.Restart.

Equipotential contours are the same as a topographic contour map you might see for mountains (as below). The contours are equally spaced so that each contour represents a given change in voltage (topographic maps have contours equally spaced for certain heights). What is the difference in voltage between contours on the equipotential surface in the animation?

Figure \(\PageIndex{1}\): A mountainous region (left) and its contour map.

Figure \(\PageIndex{2}\): Image credits: United States Geological Survey

• What is the work done on the test charge (by you) as you move it along the equipotential curve it started on?
• What is the work done (by you) as you move it toward another charge?
• The test charge is positive. If the work done by you is positive as you move it toward a charge, what sign does that charge have?

The change in potential energy is proportional to the change in electric potential (with the charge being the proportionality constant). If the change in electric potential is zero, then so is the change in potential energy, and therefore the work done is zero. As you move a positive charge toward another positive charge, you must do a positive amount of work (the charges repel and you must counteract this). As you move a positive charge toward a negative charge, you must do a negative amount of work (the charges attract and you must counteract this).

The electric field at any point is perpendicular to the equipotential contour. You can see this by looking at the force vector on the test charge as you move the charge around in this potential field. The direction of the electric field corresponds to the direction of steepest slope on a topographical map. If this were a topographical map, this would be a map of three steep mountains and a steep valley. How many positive and negative charges are on this electrostatic contour map? Note that, since the electric field is perpendicular to the equipotential lines, if we move along an equipotential, no work is done (the electric force and the displacement are perpendicular).

Illustration authored by Anne J. Cox.

Illustration 3: Electric Potential of Charged Spheres

The animation shows the equipotential contours around two charged spheres. Restart. You can change the charge of particle A by using the text box. As you click-drag your mouse around the screen, you can see the magnitude of the electric field as well as the electric potential (position is given in meters, electric potential is given in volts, and electric field strength is given in newtons/coulomb). The zero point of electric potential is infinity (far away from the charge distribution).

Change the value of the charge of \(A\) so that the charges are equal. Where (if anywhere on the screen) is the electric field zero? Where is the electric potential zero? What happens if the charges have equal and opposite charge? In this case, where is the electric field zero? Where is the electric potential zero?

What happens to the equipotentials if you make charge \(A\) more positive? What happens to the equipotentials as you make charge \(A\) more negative?

The electric potential due to one point charge is proportional to the charge divided by the distance to the charge (\(V = k q/r\)). When charge \(A\) is equal to charge \(B\) (in magnitude and sign), where do you need to put a third charge, negative but with the same magnitude of charge, in order for the potential in the middle of \(A\) and \(B\) [at point \((0,\: 0)\)] to be zero? Add a charge and move it to the correct spot to check your answer. Is there more than one place you can put this charge? The electric potential of the original two charges, as measured at the origin, is \(V = k(2Q)\), since \(r = 1\text{ m}\). The electric potential of the third charge must be \(V = -k(2Q)\) to cancel this electric potential at the origin. Therefore, the third charge must be placed at any position that is a distance of \(r = 0.5\text{ m}\) from the origin.

Illustration authored by Anne J. Cox.

Illustration 4: Conservative Forces

This animation shows the work done in moving a particle in two different force fields. As you move the particle, a vector shows you the direction of the force, and the bar and table show you the total work done as you move the particle around (position is given in meters and work is given in joules). You can zero out the work at any position you want by pushing the "set \(\text{Work }= 0\)" button. Restart.

Simply from the direction of the force (if we assume a positive test charge), if both of these fields were electrostatic, where could charges be located to produce this type of force? As you move the test charge around, notice that the force is biggest at \(y = 0\) on the right edge and points to the left. As you move the charge away from the right edge at \(y = 0\), notice that the force decreases quickly and points radially outward from a point near \(x = 10\text{ m}\), \(y = 0\text{ m}\). This lets you know that a positive charge could be located near \(x = 10\text{ m}\) and \(y = 0\text{ m}\) to approximately produce these fields.

One of these fields, however, cannot be an electrostatic force field because it is not conservative. In other words, the amount of work done depends on the path taken. If you move the particle to a particular point, it matters if you go straight there or take a circuitous route. Which force field is conservative and which is not? You can mark an initial point and an ending point on the grid, if you want to help keep track of where you've moved the particle, to compare the work done along different paths between the same two points.

Drag the particle away from an initial point and then bring it back to the same spot. How much work is done in the conservative force field? How much is done in the nonconservative force field? For which one of these forces could you get a different answer by moving the particle differently? This means that it takes a different amount of energy to bring the particle to the same spot. Could you uniquely define the potential energy function?

No, you could not. Only conservative forces can have potential energy functions since we can define the potential energy uniquely. Since the electrostatic force is a conservative force, we can develop an associated electric potential that provides easier ways to solve problems, in many instances.

Illustration authored by Anne J. Cox.
Script authored by Mario Belloni and Wolfgang Christian and modified by Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.