Potential Difference in a Uniform Field

When working with electric forces, we found it quite useful to be able to calculate the electric field at a particular location without having to worry about what kind of charged particle might come along to be affected by this field. When we know the electric field at a location, to find the electric force on whatever particle happens to be at that location, we simply multiply by the charge of the particle, q2:
Likewise, when calculating changes in potential energy of a system, it would be useful to find a quantity that is independent of the charge of the particle that might move through a region. Then we could simply multiply this quantity by the charge of whatever particle happened to move through the region, to get the change in potential energy of the system. We would like to be able to write:
In the two cases discussed in the previous section (proton or electron moving in a uniform field), many things about the system were the same: the capacitor, the electric field in the region, and the path taken by the particle between the initial location A and final location B. Let's rewrite the change in electric potential energy in each case as the product of the charge of the displaced particle and another quantity:
The quantity that is the same in both cases is (−ExΔx). This quantity is called the “difference of the electric potential between locations A and B,” and is given the symbol ΔV.
  Potential Difference

Like energy, electric potential is a scalar quantity. The dimensions of a difference in electric potential ΔV are joules/coulomb, or “difference of energy per unit charge.” This is such an important quantity that it has its own name:

Units of Field and of Energy

Note that since ΔV = −(ExΔx + EyΔy + EzΔz), and the units of ΔV are volts, then the units of electric field must be volts per meter. This is in fact equivalent to newtons per coulomb, and can be used interchangeably.

We can also now understand the origin of the energy unit “electron volt,” or eV. One eV is equal to 1.6 × 10−19J. If an electron moves through a potential difference of one volt, there is a change in the electric potential energy whose magnitude is

The electron volt is a convenient unit for measuring energies of atomic processes in physics and chemistry. Note that an electron volt is a unit of energy, not potential (potential is measured in volts, not electron volts).

Path Not Parallel to Electric Field

The path between two locations does not have to be parallel to the electric field, as shown in Figure 17.6. The general three-dimensional expression for change in electric potential in a uniform field is:
Figure 17.6    In general, ΔV = −(Ex Δx + Ey Δy + Ez Δz).

Note that this quantity can be positive or negative, since each component of the electric field can be positive or negative, and each component of the displacement can also be positive or negative.

The quantity can be written as the “dot product” of the electric field and the displacement vector :
  Potential Difference in a Uniform Field
or, using dot product notation:
example

Potential and Potential Energy

Figure 17.7 shows two locations, A and B, in a region of uniform electric field. For a path starting at A and going to B, calculate the following quantities:

(a)  
the difference in electric potential,
(b)  
the potential energy change for the system when a proton moves along this path, and
(c)  
the potential energy change for the system when an electron moves along this path.
Figure 17.7    A region of uniform electric field.

Solution  

(a)  
Potential difference:
Initial location: A
Final location: B
(b)  
For a proton:
(c)  
For an electron:

example

Field and Potential

Suppose that in a certain region of space (Figure 17.8) there is a nearly uniform electric field of magnitude 100 N/C (that is, the electric field is the same in magnitude and direction throughout this region). If you move 2 meters at an angle of 30° to this electric field, what is the change in potential?
Figure 17.8    Moving at a 30° angle to the electric field.
Solution  


17.X.5  
For a path starting at B and going to A (Figure 17.9), calculate

(a)  
the change in electric potential,
(b)  
the potential energy change for the system when a proton moves from B to A, and
(c)  
the potential energy change for the system when an electron moves from B to A.
Figure 17.9    A region of uniform electric field (Exercise 17.X.5. and Exercise 17.X.6).

Answer:

(a)  
300 V;
(b)  
4.8 × 10−17 J;
(c)  
−4.8 × 10−17 J

17.X.6  
For a path starting at B and going to C (Figure 17.9), calculate

(a)  
the change in electric potential,
(b)  
the potential energy change for the system when a proton moves from B to C, and
(c)  
the potential energy change for the system when an electron moves from B to C.

Answer:

(a)  
0;
(b)  
0;
(c)  
0



Calculating Field from Potential Difference

We have just seen how to calculate potential difference ΔV from the electric field . Conversely, we can calculate electric field from potential difference. Moving in the x direction, we have ΔV = −ExΔx, or
example

Calculating Field from Potential Difference

A capacitor has large plates that are 1 mm apart, and the potential difference from one plate to the other is 50 volts. What is the magnitude of the electric field between the plates?
Solution  
Choose the x axis to be perpendicular to the plates. Then


Gradient of Potential

In the limit of very short distance, Ex is given by a derivative:

This is called the “gradient” of the potential, by analogy with the gradient of a hill, where a steep hill might have a “15% grade.” Figure 17.10 emphasizes that a large gradient of potential (rapid change of potential with position) corresponds to a large electric field.
Figure 17.10    The negative slope of potential vs. x is the x component of the electric field.
  Electric Field is the Negative Gradient of the Potential

More technically, the gradient should be written using “partial derivatives,” like this:

The meaning of ∂V/∂x is simply “take the derivative of V with respect to x while holding y and z constant.” The fact that Ex = −∂V/∂x is closely related to the fact that (as we saw in Chapter 6 on energy) Fx = −∂U/∂x, because E = F/q and V = U/q.

The gradient is often written in the form , where the inverted delta (“nabla”) represents taking the x, y, and z partial derivatives of the potential to find the x, y, and z components of the electric field.

17.X.7  
Suppose that the potential difference between location 2.00, 3.50, 4.00 m and location 2.00, 3.52, 4.00 m is 3 volts. What is the approximate value of Ey in this region? Include the appropriate sign.

Answer:

150 V/m





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