Electric Fields and Electrical Potential EnergyElectric FieldThere is an electric field around any charged object. In other words, objects with charge (or capable of having an induced charge) brought near a charge object will experience a force.You can visualize an electric field by playing around with something called a test charge. A test charge is what it sounds like: you move it around in an electric field to “test” what the object would feel.

About a test charge: Very small (won’t alter the field) Positive (Why? Because we said so, that’s why….its just a convention, totally arbitrary) It ultimately won’t matter

We use the test charge to help us draw electric field lines. This is a way of visualizing the completely invisible electric fields.

Practice: Drawing an Electric Field

Rules for Electric Field Lines: They show the direction of the force a point charge would feel. Remember, point charges

are positive, so electrons or negative charges would experience the opposite force. The more lines there are in an area, the stronger the field is there.

Practice: Drawing the Electric Field on Multiple Charges

Story Notes, created with materials from Knapp notes

-Q

+ Q q

-Q

-Q-Q-Q-Q-Q-Q-Q-Q

Calculating Electric Field IntensityIf you want to calculate the electric field strength around an object, first consider force acting between an charge, Q, and its point charge q0.

F=kQ qo

d2

The test charge doesn’t really matter though, does it? We just used it to test the strength of the field around the charge Q. So we can divide it out.

E= Fq0

=kQ qo

q0 d2

E= Fqo

= kQd2

The units for electric field strength are N/C.

AP Practice: A really easy questionAn electron is accelerated from rest for a time of 10–9 second by a uniform electric field that exerts a force of8.0 x 10–15 newton on the electron.What is the magnitude of the electric field?(A) 8.0 × 10–24 N/C (B) 9.1 × 10–22 N/C (C) 8.0 × 10–6 N/C (D) 2.0 × 10–5 N/C (E) 5.0 × 104 N/C

Story Notes, created with materials from Knapp notes

E= Fqo

= kQd2

Memorize

AP Practice: A little bit harder now

Charges –Q and +Q are located on the x– and y–axes, respectively, each at a distance d from the origin O, asshown above.

Electrical Potential EnergyReview from MechanicsPotential energy from gravity is .

Work takes place when a force acts over a distance. The unit for work is the joule (J). A joule is a newton meter. The change in potential energy does not depend on the height, it depends on the change in height.

Story Notes, created with materials from Knapp notes

in c re a se ing ra v ita t io n a lp o te n tia le ne rgy

F g

We have a comparable situation to that of gravity when we have two opposite charges. We have a large charge Q. Placed near this charge is a small test charge q. As we increase the distance between q and Q, we have to do work to move the charge and we increase its potential energy .

Deriving the Equation for Electric PotentialWe know that potential energy is force times distance.

The force acting between the electron and the proton is:

We can plug in Coulomb’s law for the force in the potential energy equation.

So there it is. An equation for electrical potential energy:

Practice Question: Introduction to Potential EnergyAn electron in a hydrogen atom is at its lowest energy state. It is at a distance of 5.29 x 10 -11 m from the proton in the nucleus. What is its potential energy relative to the proton?

Story Notes, created with materials from Knapp notes

- -- -+ +

+ ++ +

in c rea se ine le c tr icp o ten tia le n e rg y

in c rea se ine le c tr icp o ten tia le n e rg y

F FF FE EE E

- -

U n like ch argesF orce is a ttrac tive

L ike cha rgesF orce is rep u ls iv e

Memorize This!

Important: Potential Energy and Work are scalars. When we dealt with electric field strength and force, we were dealing with vectors, so direction mattered and magnitudes needed to be added as scalars. Because potential energy and work (which we will discuss in a minute) are scalars, we do not need to worry about a direction. We can simply add their magnitudes when adding them.

AP Practice Question: Scalars and Vectors

Work in Electric FieldsChanges in the electric field = Work

Just like in a gravitational field, you do work by moving the object closer or further away.Unlike charges Pull apart +W Increase potential

Push together -W Decrease potentialLike charges Pull apart -W Decrease potential

Push together +W Increase

Deriving the Equation for WorkWe know that work is: The force exerted on a test charge in an electric field is:

Story Notes, created with materials from Knapp notes

We now put these two equations together and we get:

This gives us an equation for the work done in an electric field in moving a charge form one position to another:W is the work done in Joules, q is the charge of the test charge (not the charge making the field), E is the electric field strength, and d is the displacement of the charge.

Dealing with SignsThis is awkward. Generally, when dealing with work, we’ve thought about things from our perspective- the work we as an outside force need to do. In order to get your signs right with electrostatics, you need to think about work from the field’s perspective.

General Rule: The field does positive work a charge in order to decrease its potential energy.That means that if you use the equation above, it will give you the work the field does in moving a particle.

If you are asked about the work an outside force does, you will need to use the opposite sign.Scenario 1

Let’s talk through a scenario. Let’s say the charge in the scenario on the left is positive and it is moving from Point A to Point B.We deal with the electric field strength’s direction by putting a direction on d.

If you move in the direction of the electric field, d is positive.

So, using W=qEd, the field does positive work. The particle was moved toward a place of lower potential energy. So we can say: For this scenario, where a positive charge is moved from A to B:

q +E Deal with this with dd +W +∆U -

Story Notes, created with materials from Knapp notes

Memorize

A B

d

E

Memorize

The equation only works if you state work in terms of the work done by the field.

Story Notes, created with materials from Knapp notes

Scenario 2Now, let’s look at the same field with a negative charge.

The charge would naturally be pushed which direction? Remember that the field lines were made using a positive charge.

If the charge moved from Point A to Point B, is the field doing positive or negative work? Use the equation W=qEd

The ∆U would therefore be . So when the charge reaches Point B it has ______________ potential energy.Scenario 3Now, think about it from the perspective of an outside force. Imagine the charge is negative.

The charge would naturally want to go in which direction? Remember that the field lines were made using a positive charge.

If you (as an external force) moved the charge from Point A to Point B, would you have to do positive or negative work? Use W=qEd.

The ∆U would therefore be . So when the charge reaches Point B it has ______________ potential energy.

Potential DifferenceLet’s say we didn’t want to look at the potential energy of a specific particle, but wanted to look at the difference in potential between two given points. This is called the potential difference.

POTENTIAL DIFFERENCE CHANGE IN POTENTIAL ENERGY PER CHARGE

The symbol for potential difference is V or V.Deriving the Equation for Potential Difference

Story Notes, created with materials from Knapp notes

A B

d

E

A B

d

E

Memorize

If we look at the equation for potential energy and divide both sides by the charge q, then we get the potential difference.

We divide both sides by q (which is mathematically legal).

This equation is commonly rearranged to this form:

This is the equation that you will have available for the AP Physics Test. Notice also that the electric field can be expressed in units of V/m as well as N/C.

The unit for potential difference is the volt (V). The volt is a joule/Coulomb.

A potential difference of one volt means that it takes one joule of energy to move a one Coulomb charge in the field. A potential difference of 1000 V means that it would take 1000 J to move 1 C of charge in the field.

Potential difference is also called electric potential or voltage.

The Equations You Get on the AP Test:

Practice Question: Working with PotentialThe potential difference between two points is 12.0 V. What amount of work is needed to move a 2.00 C charge within the field?

AP Question: Work and VoltageThe work that must be done by an external agent to move a point charge of 2 mC from the origin to a point 3 maway is 5 J. What is the potential difference between the two points?Story Notes, created with materials from Knapp notes

Memorize

(A) 4 × 10–4 V (B) 10–2 V (C) 2.5 × 103 V (D) 2 × 106 V (E) 6 × 106 V

Practice Question: Electric PotentialTwo charged plates are 5.00 cm apart. The electric field between them is 775 N/C. What is (a) the potential difference between the plates and (b) what work is done moving an electron from one plate to another?

Story Notes, created with materials from Knapp notes