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5: Electric Potential

At this point, we have explored the ideas of charge, force, electric field, and electric potential energy. The last concept to tackle is the electric potential \(V\), corresponding to the gravitational potential \(U\) from our section on gravitational potential.

What is Electric Potential?

Before we begin we need to reiterate an important point from earlier: electric potential is different from electric potential energy!

Although force depends on having at least two charges, we found it useful to define a field \(\mathbf{E}\) so that we could easily determine the force on any given test charge.  Potential energy also depends on at least two charges; we can define a new quantity that takes a value everywhere to tell us something about potential energy of a test charge due to a source charge.  We call this new quantity electric potential \(V\).

The electric potential tells us how much potential energy a test charge would have at a given location due to a source charge. Specifically, the electric potential tells us how much potential energy a test charge of 1 Coulomb would have. Like energy, the electric potential is a scalar, having magnitude but no direction. Because the electric potential is defined at all points in space, it meets the qualifications of a field, and we can describe the electric potential with a scalar field.

We will define the electric potential in an analogous way to how we defined gravitational potential. 

\[PE_{\text{electric}} = q_1V_2\]

Wherein \(q_1\) is our test charge and \(V_2\) is the electrical potential created by the source charge (this operation could be done with \(V_1\) and \(q_2\) and yield the same result).

Units of Electric Potential

Knowing that electric potential energy is measured in Joules, and charge in units of Coulombs, it follows that electric potential \(V\) must have units of Joules per Coulomb. Electric potential is thus an energy density. We already have a quantity labeled \(V\) with units of J/C from Physics 7B: voltage (recall: 1 Volt = 1 J/C). As it turns out, "voltage" and "electric potential" are the same thing, and the terms will be used interchangeably from now on.

Exercise 

In the previous section, we discussed the electrical potential energy for interacting charges. In this section, we developed a relationship between \(PE_{\text{electric}}\) and the electric potential for a point charge. Determine the equation for the electric potential from a point charge.

Voltage and Potential

Our prior experiences with electric potential were primarily in circuits. In a circuit, an electron gains a certain amount of energy in traveling across a battery from low voltage to high voltage. To draw an analogy with gravitational interactions, the process is similar to being lifted vertically and gaining gravitational potential energy as it travels from low to high gravitational potential.

As we learned in Physics 7B, in an electrical circuit the electric potential (voltage) drops by \(IR\) across every resistor. Imagine we have a string of resistors hooked up to a battery as in the picture above.  The potential falling across a resistor can be modeled by the gravitational potential of a ball falling as it rolls down a ramp.

Representing V with Equipotentials

We will continue with the example of a string of resistors hooked up to a battery. If all of the resistors have the same resistance \(R\), then the voltage drop across each resistor will be equal. To represent the electric potential, or voltage, we typically draw "equipotential maps” by connecting locations with the same potential with dashed lines. Furthermore, we typically indicate only equally spaced potentials (for instance, we could choose 3V, 2V, 1V, 0V, and -1V but we would not choose 3V, 2.5V, 1V, and -3V). Because all resistors have the same value, the voltage drop across each one is equal, and the potential and each dashed line below is separated from the next by the same amount of potential .  

For the circuit diagram above, the equipotential representation adds little or no value to our prior circuit diagram representation. In circuits, we are only interested in what occurs along one dimension, the dimension of the wire. The real value of equipotentials applies to more dimensions, because the potential field exists at all points in space.

By knowing the value of the electric potential at a point, we know how much potential energy a charge would have it it were placed at that point. As we will see in a moment, information gleaned from the equipotential map also indicates the relative strength of the magnetic field, which can be used to determine the relative magnitude (and direction!) of the force on a point charge at a specific location.

As with the electric field, any configuration of charges has a unique electric potential. That is to say, the potential from a spherical charge, a capacitor, and a charged wire all have unique potentials and fields. You will have the opportunity to explore several of these during DL.

Example

  1. Draw three or four equipotentials for a proton, spanning the range between 10V and 30V. Be sure to include an appropriate scale on your equipotential map.
  2. Using your equipotential map, estimate the value of the potential at the Bohr radius, 0.529 Å, which corresponds to the electron’s average separation distance from the proton in the lowest energy state in the Hydrogen atom.
p class="boxtitle" style="visibility: visible; text-align: justify;">Solution

a) The electric potential from a point charge \(Q\) is \(kQ/r\) . We are asked to span the range of voltages between 10V and 30V with three or four potentials. We know that our equipotentials should be chosen with equal voltage differences, so we chose10V, 20V, and 330V. Before blindly plugging numbers into the formula, let’s think about what we expect to draw.

  • The potential is proportional to \(\frac{1}{r}\) , so as we move further away from the proton, the potential will decrease. Applying this idea to our problem, we expect the 30V equipotential to be closest to the proton and the 10V furthest away.
  • Furthermore, the potential decrease faster for smaller distances, such as from 1m to 2m, than it does for larger distances, such as from 99m to 100m (not convinced? Think of the graph of the function \(1/r\) or plug the numbers into your calculator). Because of this, we expect the 20V equipotential to be closer to the the 30V potential than the 10V

Now that we’ve thought through our expectations, we can look up the values of \(k\) \((9 \times 10^9 \frac{\text{Nm}^2}{\text{C}^2})\) and the charge of a proton \((1.6 \times 10^{−19} \text{ C})\) in the appendix (or on the internet). We calculate the value of \(r\) for each of the three voltages, starting with 10V: potential. \[10\text{ V} = \dfrac{(9 \times 10^9 \frac{\text{Nm}^2}{\text{C}^2})(1.6 \times 10^{-19} \text{ C})}{r_{10V}}\] \[r_{10V} =1.44 \times 10^{-10} \text{ m}\] \[1.44 \text{ m} = 1.44 \times 10^{-10} \text{ m} \times \left( \dfrac{1 Å}{10^{-10} \text{ m}} \right) \] \[r_{10V} = 1.44Å\]

Similarly for the other voltages, \[r_{20V} = 0.72  Å \] \[r_{30V} = .48  Å \]

Next, we plot the values, being sure to indicate the scale


Comparing our graph to our expectations, we find they match, so we can proceed to the second part of the problem.

b) Using our scale, we determine where the Bohr radius fits into the picture. The location is marked with a solid dot. Apparently, the value of the potential is between 20V and 30V, but much closer to 30V. We might estimate a value of 28V.

Though it is not asked for in the problem, we can calculate the value of the Bohr potential at that location. Plugging numbers into the formula, we find just over 27V. Our prediction wasn’t bad!

Relationship Between Electric Field and Potential

Thus far we have focused our attention on the relationship between the potential energy and the electric potential \(V\) . We noted earlier that both \(\mathbf{E}\) and \(V\) depend entirely on the source charges. We now explore the relationship between these quantities.

Let’s think about two interacting spherical charges one last time. We can find that the electric potential created by charge \(Q\) is (\(\frac{kQ}{r}\)). Go from the electric potential created by \(Q\) to the electric potential energy of charge \(q\), simply multiply by \(q\), as in this section (\(\frac{kqQ}{r})\). Using this, we take the derivative with respect to separation distance to find the force between the charges, as we did previously.  This gives us (\(\mathbf{F} =\frac{kqQ}{r^2}\), a familiar result.

Given the force of charge \(Q\) on charge \(q\), it is simple to recover the electric field created by\(Q\); simply divide the force by the magnitude of charge q (\(\frac{kQ}{r^2}\). In a roundabout way, we have found a relationship between electric potential and electric field, using the relationships developed in earlier sections as follows:  take \(V\) , multiply by \(q\), take the spatial derivative, divide by \(q\). This may seem overly complicated, but we can simplify by removing \(q\):  simply take the derivative of the electric potential with respect to separation distance \(r\) to obtain the same result.

What does this mean? (Refer to the calculus appendix or your favourite introductory calculus book if you need a refresher on calculus.) The electric field is large when the derivative of the electric potential is large.  The inverse is also true a fast-changing potential indicates a large field.  A large electric field corresponds to a steep slope on a graph of electrical potential \(V\) vs. position \(r\). In the equipotential map representation, a large electric field corresponds to potentials that are close together (in those locations, the potential changes rapidly over short distances).

Four Box Diagram

In a previous section, we introduced all of the relationships between force \(\mathbf{F}\), electric field \(\mathbf{E}\), electric potential \(V\), and Potential Energy \(PE\) in a diagram. This diagram is recreated below.

Screen Shot 2016-05-08 at 1.27.54 PM.png

Study the relationships again, after reading more about electrical phenomena and again after completing DL activities on the subject matter. Memorizing this diagram will not enable you to understand the physics in this section, nor to ace your quizzes. Instead, look at the diagram as an organizing structure. Ask yourself the following types of questions to become familiar with it.

  • What is common between the quantities \(PE\) and \(V\)?
  • What is common between the quantities (\matbhf{F}\) and \(PE\)?
  • How do you get from a quantity on the left of the diagram to a quantity on the right?
  • In what ways can we represent each of the quantities (what types of diagrams or graphs can you use?)
  • In a diagram of one item (such as a graph of electric potential \(V\) vs. distance \(r\)), how can we gain information about other quantities (such as \(PE\) or \(\mathbf{E}\))?