Download Chapter 23: Electric Potential

Chapter 23: Electric Potential
• Electric energy is indispensable ingredient of the our technology
All electric and electronics devices operate on electric energy
• In this chapter, we will learn the followings
• Calculation of electric potential energy of a collection of charges
• Importance of electric potential
• Electric potential (sometimes called as voltage) produced by charged particles
• Using equipotential surfaces to visualize the potential varying in a space
• Using electric potential to calculate the electric field
• Why? Benefits of learning electric potential energy and electric potential
Use of potential concept makes a lot of problems easier to answer
Concept of electric potential is crucial to understand electric circuits
Conservative Forces and Potential Energy
While any charged object moves in region where electric field exist, there will be an electric force acting on the object. Electric force may do a work on it depending on the path it follows.   b
  F  dl   Fdl cos 
b
Reminder: Work done by a force
Wa b
a
a
If a force is conservative then there should be a potential related to this force.
Also, work done by a conservative forces doesn’t depend on the path (only depends on a and b)
Similar to Gravitational forces, Electric forces are conservative forces
So there must be a electric potential energy U Reminder: Definition of potential energy
This is valid for any conservative forces
U  Wa b
  (U b  U a )  Wa b
Now, if we combine this expression with Work‐Kinetic Energy Theorem
Work‐Kinetic Energy Theorem:
Finally: K  Wa b
(U b  U a )  K b  K a
 K b  K a  Wa b
 K a + U a = Kb + U b
This is the conservation of the Mechanical Energy = K + U is conserved!
Electric Potential Energy in a Uniform Field 1/2
Let’s study the potential energy of the charged particle moving in a uniform electric field.
Using pair of metal charged plates to create uniform field and image qo positive charge moving from point a to point b as shown in the figure.
Because E. Force is a conservative force, you may choose any pathway to take your integral to calculate the work. To make it easy, we calculate the integral through the y axis from a to point b.
(Positive work)
Remember: Positive Work = Increases Kinetic Energy, K
On the other hand, it decreases Potential Energy, U
U  Wa b   (U b  U a )  Wa b
U b  U a  Wa b
 U b  U a  Wab
Electric Potential Energy in a Uniform Field 2/2
(Positive work)
U b  U a  Wa b
 U b  U a  Wab
We can derive a formula for the potential energy
for this specific example
Consider the fact that it is decreasing in downward
Choose the potential at y=0 as U= 0 (reference point)
Then consider now it only moves from a to y ( 0<=y<=a)
U b  U a  Wa b
now becomes U a  U y  qE (d  y )
This tells us that the Potential energy as a function of y
U b  U a  Wa b
now becomes U a  U y  qE (d  y )
Finally:
U y  qEy
Note: This results is very similar to gravitational potential energy U=mgh.
Conclusion: If the charge is moving in the same direction to Electrostatic Force, U decreases.
If the charge is moving in opposite direction to Electrostatic Force, U increases. Electric Potential Energy of Two Point Charges 1/2
Electric potential energy is not restricted to only uniform electric field.
We can also apply this idea to static charge distributions.
In order to apply this idea, let’s consider a charge +q (fixed) and a test charge q0 is changing it’s position with respect to +q Because the force is only radial, it only does work
while the charge q0 is moving along radial axis.
Or considering that the integral should be independent
of the path, we can do the integration only along the radial axis as shown in the figure below
Remember:  U b  U a  Wa b
Electric Potential Energy of Two Point Charges 2/2
We have driven the electric potential of two charges q and q0 as;
In order to apply this idea, let’s consider a charge +q (fixed) and a test charge q0 is changing it’s position with respect to +q Let’s look at the possible cases and try to interpret them using a graphics
Potential energy is always given with respect to a reference point. In the previous example, we have chosen ref point U=0 by choice
As you may see from the plots, the U  0 at infinity. So our ref. point is already chosen as infinity.
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Electric Potential Energy with Several Point Charges
We have driven the electric potential of two charges q and q0 as;
What if we have several charges?
In this case the electric potential energy between q0 and q1, q2, q3, …
But, this is only the energy between the q0 and other charges
We can write this equation to account for the interactions between all charges, which
gives the total potential energy of the this charge distribution:
Note: “< “ restriction is used to avoid double counting
This is the energy required(work needs to be done) to bring all this charges to this position.
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Electric Potential – Description ‐
1/2
So far, we have learned the potential energy of a charge in an electric field. The expressions, we had depends on the characteristic properties of the field and also to the charge (usually test charge)
Now, we will define “electric potential” as the electric potential energy per unit charge.
We will divide the potential energy by charge, which gives us something only depends on the field.
The concept of electric potential is very useful in calculations involving energy of charged particles
As we will see later, electric potential can be related to electric field, electric potential sometimes can be used to calculate the electric field. Eventually, this concept will lead us to basic circuit elements.
Electric Potential V (sometimes called voltage): Electrostatic energy U per unit charge.
U
V
q0
It is a scalar quantity, and SI – unit for electric potential volt ( 1 V).
Unit is named in honor of Italian electrical experimenter Alessandro Volta (1745‐1827).
1 V = 1 Volt = 1 J/C = 1 Joule / Coulomb
Potential energy of a charge q0 at electric potential V: U
 q0V
Electric Potential – Description ‐
2/2
We have learned how to calculate the potential energy by terms of work …
Dividing this by the charge moving in the field gives us the potential V  U q0
Here, Ua is the electric potential energy of q0 when it is located at point a.
Ub is the electric potential energy of q0 when it is located at point b.
by dividing the potential energy, we get the electric potential for these points.
Va is the electric potential at point a.
Vb is the electric potential at point b.
ΔU = (Ub – Ua) change in the potential energy of q0 while going from point a to point b.
ΔV = (Vb – Va) change in the potential from point a to point b. (independent from q0).
A Regular Battery: Mostly potential difference is mentioned rather than the potential energy
(+) terminal potential Va is higher than (‐) terminal potential Vb
Vab  V  (Va  Vb )  1.5V
This potential difference is often called voltage
When a charge moves from a to b it’s potential energy will change
U  q0 (Vb  Va )
Calculating Electric Potential of Point Charges
We have already found the electric potential energy of a test charge q0 as the following
Remember: Electric potential V  U q0
Potential at Point P?
Potential will be sum of the potential from each charges
‐scalar sum‐ not vector sum for the electric field. For A Continuous Charge Distribution: ‐ line – surface – volume ‐
This summation can be taken through integral –scalar sum‐
Calculating Electric Potential from Electric Field
Electric potential and electric field are closely related quantities.
We have learned how to calculate the electric field and electric potential due to charges.
If we know the charges then we can calculate any of what we need
We sometimes don’t know the charges but only the field.
In this cases, there is a way to determine the electric potential form field or vice versa.
First we will learn how to calculate the electric potential from a given electric field.
Determining Electric Potential from a given Electric Field.
Let’s assume that we want to determine the voltage between point a and another point b in a know E field
Now, imagine a test charge q0 moves from point a to point b
Force on q0 due to Field:
Work done on q0:
Remember: Finally, Potential Difference – Voltage:
Electric Potential from Electric Field
The following integral equation can be used for any electric field
If the field is constant along the way from b to a then E can be taken out of integral …
If this integral > 0  a is higher than b in potential
If this integral = 0  a is at the same potential with b If this integral < 0  a is lower than b in potential
Let’s now look at what this integral means for the field around a point charges
E and dl in the same direction
Potential decreases
E and dl in opposite directions
Potential increases
Electric Field and Potential Difference between charged Plates:
As we already know, electric field between charged plates is uniform (direction from = + to ‐)
Because V decreases in the direction of E field, + plate is higher than – plate
We can use the integral to calculate the potential difference
 plate
V  V 

 plate
 
E.dl  V  Ed
Units for Electric Field and Potential Difference
We defined the unit for Electric Field as N/C.
This tells us unit for E could be also V/m. V
U
q0
 volt = joule / coulomb
Alternative Unit for Energy (alternative to Joule)
When an electron moves in a 1V voltage – it’s potential will change (increase or decrease)
Change in the potential energy:
This energy is called as “electron volt” ( 1eV) has the same dimension with energy.
This energy unit often used in the field of electricity: 1eV = 1.602 x 10 ‐19 J
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Equipotential Surfaces
We are using electric field lines to visualize the electric field in a region.
Another way to visualize electric field graphically is the equipotential surfaces.
It is very similar to topographic maps used in geography:
By analogy to contour lines in this map, equipotential surfaces are three dimensional surfaces on which the potential field is constant all along this surface.
For a point charge, equipotential surfaces will be spherical surfaces around the charge.
Because V of a point charge only depends on r, and spherical surface represents constant r. Electric field lines are perpendicular to equipotential surfaces. Using this fact, it also helps us to visualize the electric field.
Equipotential Surfaces for Point Charges
Two dimensional plots for equipotenatial surfaces:
Important note for conductors:
When a conductor is electrostatic equilibrium, the surface of the conductor is an equipotential surface.
Because:
The electric field lines at the surface of a conductor are perpendicular to surface.
Because the equipotential surfaces are perpendicular to field lines, surface of a conductor is an equapotential surface. This tells us that the potential along the surface of conductor must be constant (same) whatever the shape or geometry it has. Calculating Electric Field from Electric Potential 1/2 Because the electric field is a vector quantity and electric potential is a scalar quantity, the electric potential calculations are easier most of the times.
So, is there any way to calculate the electric field from an electric potential?
In order to answer this question, let’s look at the relation between electric field and potential:
Remember: inverse operator for integral is the derivative. Because the electric field has three components in three dimensional space, solution is the derivative of potential with three components (this is called as gradient operator).
: Negative of the gradient of potential gives us the electric field.
Gradient of a function in Cartesian coordinates (x,y,z):
Calculating Electric Field from Electric Potential 1/2 : Negative of the gradient of potential gives us the electric field.
Gradient of a function in Cartesian coordinates: when V(x,y,z) Gradient of a function for a Spherically Symmetric Potential : when V(r) Radial component of the electric field ‐ only radial ‐

V
E
rˆ
r
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RECOMMENDED END OF CHAPTER‐23
QUESTIONS AND PROBLEMS
1,4,5,6,7,8,10,12,14,15,17, and 21
1,2,3,4,5,8,9,10,11,13,14,15,16,18,19,21,23,25,27,
28,29,30,32,34,35,36,38,39,42,43,44,45,46, and 47
50,51,56,59,62,68,70,73, and 86