Electric FieldsChapter 21

E = F/Q

Electric Field• The electric force is a field force, it applies force

without touching (like the gravitational force).• In the region around a charged object, an Electric

Field is said to exist.• An electric field exists in a region if space if a charge

placed in that region experiences an electric force.• The magnitude of an electric field at any given point The magnitude of an electric field at any given point

is defined to be the ratio of the force on a charge at is defined to be the ratio of the force on a charge at that point to the amount of charge.that point to the amount of charge.

E = F/Q• Electric field strength has units of Newtons/Coulomb

(N/C).

Electric field equation for a point charge

• E is electric field strength, units: N/C • E is a VECTOR• q0 is a + test charge, units: C• q is the charge producing the field, units: C• r is the distance between charges, Units: m• FE is the Electric Force, units: N• kc is the coulomb constant, 8.99x109Nm2/C2

0

EFE

q

����������������������������

becomes 2c

qE k

r for a point

charge

E-Field vs g-field

0

0

q

FE

FieldE

0m

Fg

fieldg

g

E-Field Calculus

204

1

r

dQEd

Electric Fields• Electric fields are represented by electric field lines

(imaginary) and show the direction of the force at that point.• Electric field direction is from a positive charge to a negative

charge.• Electric field lines (also known as electric flux) do not touch

each other.• Electric field line represents the path a small positive charge

would move in the electric field.

Rules for Drawing Electric Field Lines1. The lines must originate on a positive charge (or

infinity) and end on a negative charge (or infinity).

2. The number of lines drawn leaving a positive charge or approaching a negative charge is proportional to the magnitude of the charge.

3. The relative density of the field lines represents the magnitude of the field (where lines are closer together the field is stronger).

4. No two field lines can cross each other.5. The line must be perpendicular to the surface of

the charge

Field lines for a single positive and a single negative charge

+ -

http://www.physicslessons.com/exp21b.htm

Field lines for a positive and negative charge

The field is stronger where the lines are closer together

Field lines for two equal positive charges

+ +X

The point X in the diagramis a neutral point.There is no electric field at point X.

Field lines for two equal negative charges

- -X

The point X in the diagramis a neutral point.There is no electric field at point X.

Electric Field between two charged plates

+ + + + + + + +

- - - - - - - -

Predict the sign of the charges pictured below and the relative charge at each point. Justify your answer.

Two charges are placed on the x axis. The first, with a charge of +Q, is at the origin. The second, with a charge of -2Q, is at x = 1.00 m. Where on the x axis is the electric field equal to zero?

• This question involves an important concept that we haven't discussed yet: the field from a collection of charges is simply the vector sum of the fields from the individual charges. To find the places where the field is zero, simply add the field from the first charge to that of the second charge and see where they cancel each other out.

• The correct solution is x = 2.41 m

Gravitational Potential• Lift an object into the air and we give it

gravitational potential energy

PE Work

PE Force distance

PE mg h

PE mgh

Electric Potential

• Separate a positive charge from a negative charge and give it electric potential energy

PE work

PE force distance

PE Eq h

PE Eqh

• Electric potential is defined as the electric potential energy per unit charge at a specified location.

• The units for volt, V• The electric potential at a point a distance r from a

point charge Q is given by: V = k Q / r • The total potential at a point P is the sum of the

potentials due to individual point charges present.

Electric Potential

Electric Potential=electric potential energy

amount of charge

1 Volt =1 Joule

CoulombV = EPE/Q

Equipotential lines for a point charge, and for a constant field

No work is required to move a charge along equipotential lines.

Electric Potential Difference• The electric potential difference is the difference

in electric potential (V) between the final and the initial location when work is done upon a charge to change its potential energy.V = EPE/Q or V = WonQ/Q

• One VOLT is the electric potential difference between two points when one Joule of work is done in moving one Coulomb of charge between the points.

• The SI unit of electric potential difference is the VOLT, named in honor of Alessandro Volta.

• Because electric potential difference is expressed in units of volts, it is sometimes referred to as the voltage.

Electric Potential in a Uniform Field• In the uniform field between two

charged plates there is a constant force on a charged particle. E = F/Q so F=EQ

• Work done by the uniform electric field W=Fx=EQx since V = WonQ/Q then V=EQx/Q = Ex so

x

V = Ex or V = Ed

x and d in the equation represent the distance the charge moves, and E is the electric field

Electric Potential in a Uniform Field-Graphs

+qF = qE

d

Electric Potential DifferenceV= E-field x distance

[V] = Nm/C= J/C= Volt

Potential increases as moving against E-field.

Work done to unit charge.

E

V

F

0 d

Electric Potential in a Uniform Field – different chargesE

+qF = qE

d

-q

High potentialLow potential

A B

Potential energy for +q increases A -> B.

However, potential energy for –q increases B -> A.

Electric potential at B is higher than at A for both.

What if the movement is not parallel to the field?

Electric Potential in a Uniform Field

+q

d

What if the movement is not parallel to the field?How much of work has been done on the charge?

W = qE dcos()

E

Electric potential changed by VV = E d cos().

Important!!Motion perpendicular to E-field does not change potential.

Electric Field near a conductor• A conductor is in electrostatic equilibrium

when the charge distribution (the way the charge is distributed over the conductor) is fixed. Basically, when you charge a conductor the charge spreads itself out. At equilibrium, the charge and electric field follow these guidelines: – the excess charge lies only at the surface of

the conductor – the electric field is zero within the solid part of

the conductor – the electric field at the surface of the

conductor is perpendicular to the surface – charge accumulates, and the field is strongest,

on pointy parts of the conductor

E Compared to V near a charged conductor-graphs

E (the field) is zero inside but V (the potential) is not zero inside the sphere, it is constant.

Gauss's Law• The total of the electric flux out of a closed surface is equal to the

charge enclosed divided by the permittivity.

• The electric flux through an area is defined as the electric field multiplied by the area of the surface projected in a plane perpendicular to the field. Gauss's Law is a general law applying to any closed surface.

• It is an important tool since it permits the assessment of the amount of enclosed charge by mapping the field on a surface outside the charge distribution.

• Gauss' Law, Integral Form:

Applications of Gauss' Law

• Gauss' law is a powerful tool for the calculation of electric fields when they originate from charge distributions of sufficient symmetry to apply it.

Millikan’s Oil-Drop Experiment – charge on an electron• Millikan, with the help of his graduate student H. Fletcher, was

able to show that the charges of the droplets of oil falling between two parallel charged plates always carried a whole number multiple of a basic charge, qe = 1.592 x 10-19 C. Today, the accepted value for the fundamental unit of charge is e = 1.602 x 10-19 C.

• He measured the charge on the oil drops by using a fine mist of oil sprayed from an atomizer. Most of the droplets get negatively charged as they picked up some small, unknown number of electrons as they passed through its nozzle. Some of the drops then fell through a hole in the top plate and drifted into the region between the two parallel plates between which a variable electric field had been established.

• The drops where lit with an intense light and viewed through a telescope. Once a drop was located, the voltage was varied controlling the electric field, E, to slow down the drop's descent. When the drop reached terminal velocity (mg = qE) it was tracked through the remainder of its fall and a ratio of mass per unit charge was recorded. Once mass was determined, charge on the drop could be calculated.

Same Charge, Different Voltage

A B

insulating stand

A B

A B

insulating stand

A B

Same Q

Low VHigh V

Same V

Different Q

Same Q

Before touching After touching

Same V

Before touching After touching

Charge moves to lower potential

Storing Charge: The Parallel Plate Capacitor• A capacitor is a device for storing charge. It is usually made up of two plates

separated by a thin insulating material known as the dielectric. One plate of the capacitor is positively charged, while the other has negative charge.

• The charge stored in a capacitor is proportional to the potential difference between the two plates. For a capacitor with charge Q on the positive plate and -Q on the negative plate, the charge is proportional to the potential: If C is the capacitance, Q = CV. So the capacitance can be written as C = Q/V.

• The capacitance is a measure of the amount of charge a capacitor can store. It depends on area of the plates, A, and distance between them, d. Capacitance has units of farads (F).

• The energy stored in a capacitor is the same as the work needed to build up the charge on the plates. As the charge increases, the harder it is to add more. Potential energy is the charge multiplied by the potential, and as the charge builds up the potential does too. If the potential difference between the two plates is V at the end of the process, and 0 at the start, the average potential is V / 2. Multiplying this average potential by the charge gives the potential energy : PE = 1/2 Q V.

• Substituting in for Q, Q = CV, gives: U = 1/2 C V2 , where U is the energy stored, C is the capacitance and V is the voltage.

Energy Stored: The Capacitor

Another explanation for the 1/2 factor??

The reason is that in charging a capacitor, the first bit of charge is transferred while there is very little voltage on the capacitor (recall that the charge separation creates the voltage!). Only the last bit of charge is moved across the full voltage. Thus, on average, the full charge moves across only half the voltage!

Example: Parallel Plate Capacitor

• Consider a parallel plate capacitor made from two plates each 5 cm x 5 cm separated by 2 mm with vacuum in between. What is the capacitance of this capacitor?

• If a power supply puts 20 volts across this capacitor, what is the amount of charged stored by this capacitor?

Example: Parallel Plate CapacitorThe capacitance depends on , A, k and d:

Cparallel plate = A/dwhere = 1 for vacuum, A = 5 cm x 5 cm = 25 cm2 = 25 x 10-4 m2, d = 2 mm = 2 x 10-3 m, and = 8.85 x 10-12 C2/Nm2 , so C = (1)( 8.85 x 10-12 C2/Nm2)(25 x 10-4 m2)/ 2 x 10-3 m = 1.10 x 10-11 F = 11 pF .

If we have a 20 volts across the capacitor, then to calculate the charge, Q, we can use: C = Q/V to get: Q = C*V = (11 x 10-12 F )(20 V) =

2.2 x 10-10 Coul = 0.22 nC = 220 pC.

Capacitors-Graphs

-

-- -

- -- - -

- --

--

-

+Q

-Q

V

Q

Q = C V, so C is the slope

V =

V = EPE/Q or EPE = VQ if EPE = W then W = QV so work is the area under the curve

or electromotive force (emf) sometimes used instead of V for voltage source

Other types of capacitors

• Note: We can have other shapes for capacitors. These other shapes will have formulas for them that differ from the formula for parallel plates. These formulas will also show that the capacitance depends on the materials and shape of the capacitor.

Electron-Volts

• Another unit of energy that is commonly used in atomic and nuclear physics is the electron-volt.

• One electron-volt is defined as the energy a charge-field system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt– 1 eV = 1.60 x 10-19 J

Electric/Gravity Comparison

GravitationalGravitational

(on the earth)(on the earth)ElectricElectric

Mass, Mass, mm (Kg) (Kg)

Only 1 typeOnly 1 typeCharge, Charge, QQ (C) (C)

++ and and --

g (m/sg (m/s22=N/Kg) =N/Kg) E-field (N/C)E-field (N/C)

mmgh (Nm = J)gh (Nm = J)Potential energyPotential energy

QQEd (J)Ed (J)Potential energyPotential energy

gh (Nm/Kg) gh (Nm/Kg) Gravitational potentialGravitational potential

Ed Ed Nm/C)Nm/C)Electric potentialElectric potential

Sources• Physics by Zitzewitz• www.glenbrook.k12.il.us/GBSSCI/PHYS/Class• http://webphysics.davidson.edu/course_material/

py230l_wc/demo/illustration22_1.html http://members.aol.com/physicsfirst/

• http://physics.bu.edu/~duffy/PY106/Charge.html• http://physics.bu.edu/~duffy/PY106/Electricfield.htm

l• http://dev.physicslab.org/Document.aspx?

doctype=3&filename=AtomicNuclear_MillikanOilDrop.xml