D2d HL only - Electric potential energy and Equipotentials

• Physics
• Secondary Education

Coulomb's law and electric fields are fundamental concepts in understanding electricity and magnetism.

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D2c HL only - Electric potential energy and Equipotentials

Objectives

• Define Electric Potential Energy
• Compare EPE to GPE
• Define equipotentials and their relation to field lines 

What are some key points you remember about gravitational potential energy?

Recap: the Gravitational field

A gravitational field is similar to an electric field because it also an action-at-distance force.

This means that the force from the source can be felt without making contact with the source. 

The source of the gravitational field is usually a very large mass (e.g. a planet)

Gravitational potential difference

When you lift a mass a certain height above the Earth you have increased its potential.

The potential difference between two points in a gravitational field is expressed in Joules.

This is because it is the energy difference between these two points.

Gravitational potential difference

Work done is also a unit of energy.

Thus the energy needed to move a mass from a low potential region to a higher potential is also known as the work done to move the mass between these two regions in the gravitational field.

In a gravitational field we usually assume the ground is at zero potential.

To lift an object of mass m a height h in a uniform gravitational field g without acceleration, you must apply a force mg. The work you do is +mgh, while the work done by the field is - mgh. When you lower the object, you do negative work and the field does positive work.

Near the surface of a negatively charged object, the electric field is nearly uniform.

To lift without acceleration a positive charge q in a downward field E requires a force qE. You do positive work in lifting the charge, and the field does negative work. The signs reverse when you lower the charge. 

The work your applied force does on the mass or on the charge can go into kinetic energy, waste heat, or potential energy.

If there is no friction and no acceleration, then the work you do goes into a change of potential energy: ΔU = mgΔh for a mass in a gravitational field and ΔU = qEΔh for a charge in a uniform electric field. The sign of Δh determines the sign of ΔU. (If a charged object is moved in a vicinity where both types of fields are present, we’d have to use both formulae).

Whether or not there is friction or acceleration, it is always the case that the work done by the field is the opposite of the change in potential energy: Wfield = - ΔU.

video placeholder - WD in an E field

A positive charge is displaced to the right by some applied force in the uniform E field displayed in the image.

What is the sign of the work done by the field?

1. positive
2. negative

A positive charge is displaced to the right by some applied force in the uniform E field displayed in the image.

What is the sign of the work done by the force?

1. positive
2. negative

Potential

Gravitational potential is defined to be gravitational potential energy per unit mass.

At any given height above Earth’s surface, the gravitational potential is a constant since                                          

Thus potential is independent of mass. If M > m and they’re at the same height, M has more potential energy than m, but they are at the same potential.

​m​​U​​=​m​​mgh​​=gh

Electric Potential

Similarly, electric potential, V, is defined to be electric potential energy per unit charge. At any given distance from a charged surface in a uniform field, the electric potential is a constant since:

Thus potential is independent of charge. If Q > q and they’re the same distance from the surface, Q has more potential energy than q, but they are at the same potential. 

In a uniform field V = E d.

​q​​U​​=​q​​qEd​​=Ed

Units for potential

Key Definitions

Electric potential energy - the total work done by an external agent (force) in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.

Electric Potential - The electric potential energy PER UNIT charge

Why is the accelertion relevant?

If there is an acceleration then there must be a resultant force. The way we defined the potential energy was by considering a force doing work on a charge in an electric field and equating the force appied to the force on the charge from the field. If one force is bigger you can't equate them.

Equipotential surfaces

Gravitational

As with gravitational potential energy, the reference point for electric potential energy, and hence potential, is arbitrary. Usually what matters is a change in potential, so we just pick a convenient place to call potential energy zero. 

The horizontal lines on the left represent equipotential surfaces--planes in which masses all have the same potential, regardless of the mass. 

On the 30 J/kg surface, for example, every kilogram of every mass has 30 J of potential energy. Note that equipotentials are always perpendicular to field lines.

Electric

The equipotentials on the right are labeled in volts. Potential decreases with distance from a positively charged surface since a positive charge loses potential energy as it recedes from the surface. Here again the equipotentials are perpendicular to the field lines. On the -45 V surface, every coulomb of charges has -45 J of potential energy. 

A -2 C charge there has a potential energy of +90 J.

Why 0V?

Remember that we can choose our 0 to be anywhere (it is arbitrary). In the IB we will always have it at infinity but here we choose it to be at the surface to make the analogy with gravity common.

Equipotential Surfaces: Positive Point Charge

Imagine a positive test charge, q, approaching an isolated, positive, point-like field charge, Q. 

 The closer q approaches, the more potential energy it has. So, potential (V) increases as distance (r) decreases.

1. A
2. B
3. C
4. D

1. A
2. B
3. C
4. D

Equipotential Surfaces: Positive Point Charge

It can be shown that the potential (V) for a point charge is given by:

This shows that V is proportional to Q, that 

 V → 0 as r → ∞, and that V → ∞ as r → 0. Equipotential surfaces are always perpendicular to the field lines, for any charge configuration.

V=​r​​kQ​​

Using the equation how would the potential change for an isolalted negative charge?

A hydrogen atom consists of 1 electron and 1 proton. The radius of a hydrogen atom 53pm. Calculate:
a. The force of attraction between the electron and proton
b. The electric field strength at the electrons orbit
c. the electric potential at the electrons orbit.

Write down 2 things you learned in this lesson.

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Ask 1 question about something you haven't quite understood yet.

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Write down 2 things you want to know more about.

Here, students enter two things they would like to know more about. This not only increases involvement, but also gives them more ownership.