Your Conceptual Goals for this section: Electric Potential (ΔV)

Connection back to Work/Energy:

You should be duly suspicious that any description that has potential in it's name will have something to do with energy. What we found in PH211 (Newtonian Physics) was that understanding energy gave us a different way to look at problems. Energy methods offered different insights and sometimes helped simplify and clarify our understanding.

The core physics concept was that forces on objects that move are do work (move energy). When we consider the work done by the net force acting on an object we get the Work/Energy theorem:

W_net = ΔKE

If we are considering individual forces we learned that the energy moved by a particular force was given by:

W_F= F ^. Δ x

....remembering that the dot product is a particular form of vector 'multiplication'. We calculate this by noting the the dot product introduces the cosine of the angle between the force vector (F) and the displacement vector (Δ x).

W_F= F cos(θ) Δ x

Any charge which is in an E field experiences forces and when the charge moves or is moved then work is done (energy is moved as well).

The last connection to keep track of is the relationship between the change is some potential, or stored, energy and the work done by a force. Remember that the difference between work done and the change in potential energy have the same magnitude but different signs because energy taken away from an object (-) ends up adding to (+) a stored energy reservoir somewhere. Formally we describe this as:

-W_AB = ΔPE_AB

In words this says the work done by a particular force as an object moves from A -> B is the opposite sign but same magnitude as the change in the potential energy stored as the object moves from A -> B. What do you expect to happen as the object moves from B -> A with the same force?

Caveat: Only forces where W_AB does NOT depend on the path (how you travelled from A -> B) can be described with potential energy.

Energy Tool(s): Our primary tool for approaching problems where energy is a factor is the energy bar charts. If you need a rough review see the discussion in the Work/Energy Breadcrumbs from last term. Bar charts are a general tool for physics quantities that are conserved. This includes energy as well as momentum and some other concepts we haven't discussed.

Gravitational Potential Differences (operational definition):

Remember how we determined the gravitational potential energy? We figured out the work done by the force of gravity as a mass is moved from point A to point B. There may well be other forces doing work as well but we were only focused on the force of gravity.

Here's the thing we didn't do with gravity. What would happen if instead of calculating the total energy stored by gravity we calculated the energy stored per unit of mass (kg)? Calculate the change in graviational potential energy when I move a 2 kg, 3.5 kg, and 53 kg object on top of a 1.3 m high counter? Now calculate the energy per kg in each case. What you have done is:

something = W_{force of gravity}/m = constant? (what are the units?)

The fact that this number is constant is of interest. It's not energy exactly but it is closely related to energy. Eventually we came to call this the gravitational potential difference between the two points. Notice how the use of language invites confusion. If I tell you that two points have a gravitational potential difference of 34 m²/s² can you tell me if they are further apart than the 1.3 m of the counter? How much energy does it take to put 1 kg up on this new location?

Notice that the change in gravitational potential is NOT the same as the change in gravitational potential energy!!!

Electrical Potential Energy (EPE):

The first thing we have to do is convince ourselves that if I figure out the work needed to move a charge from A -> B that there are many paths by which I can do that and still get the same number. This is a characteristic that is needed to be able to define a potential energy. Sketch a uniform E field, you pick the intensity, pointing in some direction. Pick two points A and B in that E field. Put a charge, you pick the sign and magnitude of the charge, at Point A. Calculate the force (direction and magnitude) on the charge. Draw a path from A to B where the first leg is perpendicular to the E field and the second is parallel to the E field and calculate the work done by F_E on each leg (one of them should be 0, why?) Can you find other paths where the work done by F_E is the same? Many other paths?

This means there is a well behaved electric potential energy like that for gravity.

Electrical Potential Difference:

Now calculate the change in the electric potential (this is completely analogous to what we just did with gravity).

change in electric potential = W_AB/q_moved

What are the units of this quantity? Is it energy? Is it related to energy?

So what does physics typically do when it has a new concept and a new set of units? It gives them a new name of course? J/C we call Volts and the change in the electric potential we label ΔV_AB (it's nice that the label and the units match eh?)

ΔV_AB= W_AB/q

Because W_AB is a scalar and not a vector that means ΔV_AB is also a scalar and NOT a vector. This is important when we get to problems with multiple charges and distributions of charges. Do you notice that this means that the change in the electric potential energy is given by

-ΔPE_E= qΔV_AB= W_AB

In our energy bar charts we often label this as qΔV potential energy.

Electric Potential (implied reference point):

In the case of gravity we didn't pursue this much because the gravitional field everywhere on the earth is very uniform and so the simpler approach of ΔPE_g = mg(h_f-h_o) works pretty well. In electrostatics, the settings we are interested in are more complex so we need to dig a little deeper.

Lets consider a charge in the space near a point charge which creates an E field.

This is a useful image -- we'll label where the charge is now A and the point P as B. Will the force on the charge be constant as it moves from A to B? How do we determine work when the force is not constant?

Remember this?

In this example they don't clearly note the dot product inside the integral but in our case what is cos(θ) between the force and the direction of displacement? We could do this by plotting the force as a function of the radial distance and then using manual techniques like we did for the Bungee II lab. On the other hand we're practicing integration so let's go do it.

Here's the place we're going (and we'll see if we get there:))....

ΔV_r = k q_src/r

In this expression there is an implicit assumption that this is a potential difference between a point r away from the source charge and a point infinitely far away. Most texts don't keep the Δ in front of the V but I think it's a helpful reminder that this is actually still a potential difference. There is no such thing as a potential but only differences relative to some reference point.

Notice also that + charges are like hills and the potential near them is higher than points far away. Source charges that are - are like holes in that the potential near them is lower than points far away. This is a new conceptual feature that doesn't happen with gravity because we don't appear to have both + and - masses.

Voltage in our houses and lives:

It seems incredibly likely that you have noticed that this unit of electric potential is the same one that we use to describe car batteries, wall outlets, and other stuff of an electrical nature. We will explore this more when we get to circuits. For now it is probably most useful to notice the conceptual connection between voltage differences and electric fields which push or pull on charges. Voltages indicate the existence of source charges and E fields which can then be used to move charges or charged objects around. A perfect example is in an ink jet printer where voltages are used to create fields that direct the charged droplets of ink to the right place on the paper to produce the font that you have selected for your term paper.

We have worked our way from electrostatic forces to E fields to ΔV over the last three weeks. From an engineering perspective we usually do it exactly the opposite way. We design the voltage to produce the force we want, overlooking the mysterious E fields that are produced, to create the effect we want in the world.

Voltages and metal objects:

Consider a solid lump of metal on which I place some excess charge. Where does that charge go? Will there be an E field inside the lump of metal? If there is no E field inside how much work does it take to move a charge from one side of the lump to the other? What does that mean about ΔV_AB between any two points of the metal? Do I change anything about the distibution of the charges if I use an ice cream scoop to take out the inside of the lump once the charges have stabilized? Why does this suggest that you are relatively safe in a thunderstorm if you stay in your car or truck?

If I have a spherical metal sphere which has some excess charge how can I determine what it's voltage is? Perhaps I will get a chance to do this as an integration problem. If not the result is that a spherical distribution of charge can be treated as if it is concentrated at the center of the sphere for all points in space outside of the spherical distribution. This is a very handy result.

Calculation: What is the potential of a 12 cm metal sphere on which 12 nC of excess negative charge has been placed?

Multiple Charges:

So what do we do with multiple charges? Because ΔV_r is a scalar we can just add up the contribution to the potential from each of the charges involved. We do have to keep track of whether the source charge is + or -.

When you solve these problems it is still valuable to make a sketch so you get the geometry right but there is no vector stuff to worry about. + source charges add to the potential at a point and - source charges reduce or subtract from the total electric potential at a point.

Integration:

Just like Coulomb Force and E Field problems in the previous sections EXCEPT that ΔV_AB is a scalar and not a vector. That's very nice!! Follow the integration frame and remember to be careful about the sign of your source charges. This is a little different than the vector problems we have been doing previously. Practice, practice, practice....