Fields Tutorial 5 - Electric Potential and Energy

 

Electric Potential, V

When we looked at gravity fields, we saw that gravitational potential energy is the energy needed to bring an object from infinity to a certain point.  We also saw that gravitational potential was the energy per unit mass.  There are two similar quantities in electric fields, potential energy and electric potential.

 

Suppose we moved a charge form one point to another in an electric field.  We would have to do a job of work against the field (or get work out if it were with the field), so there is an energy change.  We can show this as a graph:

 

 

The area under the graph represents the energy transformed.  Using the mathematical trick of calculus, we can show that the energy is given by the equation:

 

 

 

Question 1

A positive charge of 6 nC attracts a negative charge of 5 nC from a distance of 4 mm.  What is the work done?  What does the sign mean? 

Answer

 

We can define the electrical potential as the energy per unit charge.  It has the code Ve and is given by the equation:

 

 

The units are Joules per Coulomb (JC-1) or Volts (V).  Indeed the volt is defined as the energy transformed when unit charge is moved between two points.  We can show the relationship between the electric field and potential in this graph.  Potential is the area under the graph:

 

 

All points within an electric field that have the same potential are called equipotentials, rather like contours on a map.  In a radial field, the equipotentials are concentric circles.  In a uniform field, they are parallel lines equally spaced.

 

Suppose we have the positive plate in a uniform field at +300 V, and the negative plate is at 0 V.  Suppose we have 10 equipotentials equally spaced in the uniform field.  Each equipotential has a potential difference of + 30 V.  So there are equipotentials at 0 V, 30 V, 60 V, 90 V, etc.

 

Question 2

Why are equipotentials not equally spaced in a radial field?

Answer

 

 

Worked Example

If a hydrogen atom consists of a proton and an electron at an average separation of 5.0 × 10-11 m, what is the electric potential at this distance and the potential energy of the electron?  What would the field strength be?

Use Ve = __1__ Q = 9 × 109 × 1.6 × 10-19 = 28.8 V

             4pe0   r             5.0 × 10-11

Potential energy = charge × potential = 28.8 V × 1.6 ×10-19 C = 4.6 × 10-18 J

Field Strength = potential ÷ separation = 28.8 ÷ 5.0 × 10-11 = 5.76 × 1011 N/C

 

 

Question 3

A tiny negatively charged oil drop is held stationary in an electric field between two horizontal parallel plated as shown. 

The mass is 2.5 ´ 10-10 kg.

 

 

(a)    What are the two forces acting on the drop and in which direction do they act?

(b)   The drop is stationary.  What can be said about the two forces?

(c)    What is the charge on the oil drop?

Answer

 

 

Trajectories of Particles in Electric Fields

The key points to remember are:

Other charged particles have whole number multiples of 1.6 ´ 10-19 C.

 

Electrons can be accelerated by an electric field.  This happens in any cathode ray tube, e.g. a TV set, or a VDU.  Electrons are “boiled off” a red hot cathode in a process called thermionic emission.  They are then attracted by a positive electrode, the anode, at a high voltage.  This makes them move forward at a high speed.  The electrons move parallel to the direction of the field.  Most hit the anode, but some fly through a hole at the front, to hit a screen at the far end of the tube.  The screen is covered in phosphor, and the energy the electrons have is converted to light (and some X-rays).  Because this arrangement spits out electrons, it is called an electron gun.

 

 

The electrons are repelled by the cathode and accelerated by the anode.  Each electron is given energy, which can be found using energy = charge × voltage, E = QV.  The energy is entirely kinetic energy, so we can say that:

 

            QV = Ek = ½ mv2

           

            Ž v2 = 2QV

                          m

 

There is no reason why positive charges cannot be accelerated in the same way.  Indeed in a mass spectrometer, positive ions are accelerated by an electric field and bent by a magnetic field to hit a detector.  Particle accelerators work in the same way.

 

Let us consider how an electron behaves as it enters a uniform electric field at right angles:

 

 

  • We can apply Newton II to help us to work out the acceleration.  If we wanted to know the upwards velocity, we would work out the time interval, and then use the equation a = Dv/Dt to calculate the upwards velocity.  Since the electrons are in a vacuum, the horizontal velocity is not affected.

  • Once we know the upwards velocity, then we can do a vector addition to tell us the resultant velocity.

Question 4

How would the trajectory of a proton be different?   

Answer
Question 5

Two parallel plates are set at a distance of 12 mm apart in a vacuum.  The plates are 30 mm long.  The top plate has a potential of +300 V and the bottom has a potential of – 300 V as shown.  The electrons have been accelerated to a horizontal velocity of 2 ´ 107 m/s just as they enter the electric field.

 

(Electronic charge = 1.6 ´ 10-19C; mass of an electron = 9.11 ´ 10-31 kg)

 

(a)    Sketch on a copy of the diagram the electric field between the plates

(b)   What is the electric field strength at a point midway between the plates?

(c)    What is the force on an electron at this point?

(d)   What is the resulting acceleration?

(e)    What is the vertical velocity as the electron leaves the plates?

(f)     What is the resultant velocity?

Answer


Comparing Electric and Gravitational Fields

There are many analogies that can be drawn between electric fields and gravity fields.  Theoretical physicists would go as far as saying that the two are possibly different manifestations of the same thing.  Let us compare the two:

 

Feature

Electric Field

Gravity Field

Quantity susceptible to the force

Charge

Mass

Constant of Proportionality

__1__

 4pe0     

where e0 is the permittivity of free space.  The value of e can be changed by adding a material.

G

 

The value of G, the universal gravity constant has the same value for all media, including a vacuum. 

Relationship with distance

Inversely proportional to r2.

Inversely proportional to r2.

Force Equation

F =  __1__ Q1Q2

         4pe0   r2

F = -G m1m2

              r2

Direction of force

Can be attractive or repulsive

Always attractive

Relative strength

Strong at close range

Weak.  Can only be felt with massive objects

Range

Infinite

Infinite

 

The gravitational attraction between particles in an atom is so small as to be negligible.  The nucleus and its electrons are held together entirely by electrostatic forces, and these are involved in chemical reactions.

 

Gravity forces hold planets together and hold them in their orbits.  Electrostatic forces over the interplanetary distances can be ignored.