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20.4 Obtaining Electric Field from Electric Potential Assume, to start, that E has only an x

component

Similar statements would apply to the y and z components

Equipotential surfaces must always be perpendicular to the electric field lines passing through them

x x

dVd becomes E dx and E

dx E s

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For Three Dimensions In general, the electric potential is a

function of all three dimensions Given V (x, y, z) you can find Ex, Ey and

Ez as partial derivatives

x y z

V V VE E E

x y z

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Electric Field and Potential of a Dipole The equipotential lines

are the dashed blue lines

The electric field lines are the brown lines

The equipotential lines are everywhere perpendicular to the field lines

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20.5 Electric Potential for a Continuous Charge Distribution Consider a small

charge element dq Treat it as a point

charge The potential at

some point due to this charge element is

e

dqdV k

r

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V for a Continuous Charge Distribution, cont

To find the total potential, you need to integrate to include the contributions from all the elements

This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions

e

dqV k

r

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V for a Uniformly Charged Sphere A solid sphere of

radius R and total charge Q

For r > R, For r < R,

rQ

kV e

2 23

2

2

2

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eD C

eD

k QV V R r

R

k Q rV

R R

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V for a Uniformly Charged Sphere, Graph The curve for VD is

for the potential inside the curve It is parabolic It joins smoothly with

the curve for VB

The curve for VB is for the potential outside the sphere It is a hyperbola

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20.6 V Due to a Charged Conductor Consider two points on

the surface of the charged conductor as shown

is always perpendicular to to the displacement

Therefore, = 0 Therefore, the potential

difference between A and B is also zero

E

dE s ds

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V Due to a Charged Conductor, cont V is constant everywhere on the surface of a

charged conductor in equilibrium V = 0 between any two points on the surface

The surface of any charged conductor in electrostatic equilibrium is an equipotential surface

Because the electric field is zero inside the conductor, we conclude that the electric potential is constant everywhere inside the conductor and equal to the value at the surface

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E and V of a sphere conductor The electric potential is a

function of r The electric field is a function

of r2

The effect of a charge on the space surrounding it

The charge sets up a vector electric field which is related to the force

The charge sets up a scalar potential which is related to the energy

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Two charged sphere conductors connected by a conducting wire

The charge density is high where the radius of curvature is small

And low where the radius of curvature is large

The electric field is large near the convex points having small radii of curvature and reaches very high values at sharp points

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Cavity in a Conductor Assume an

irregularly shaped cavity is inside a conductor

Assume no charges are inside the cavity

The electric field inside the conductor is must be zero

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Cavity in a Conductor, cont The electric field inside does not

depend on the charge distribution on the outside surface of the conductor

For all paths between A and B,

A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity

0B AV V d E s

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20.7 Capacitors Capacitors are devices that store

electric charge The capacitor is the first example of a

circuit element A circuit generally consists of a number of

electrical components (called circuit elements) connected together by conducting wires forming one or more closed loops

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Makeup of a Capacitor A capacitor consists of two

conductors When the conductors are

charged, they carry charges of equal magnitude and opposite directions

A potential difference exists between the conductors due to the charge

The capacitor stores charge

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Definition of Capacitance The capacitance, C, of a capacitor is

defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors

The SI unit of capacitance is a farad (F)

QC

V

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More About Capacitance Capacitance will always be a positive quantity The capacitance of a given capacitor is

constant The capacitance is a measure of the

capacitor’s ability to store charge The Farad is a large unit, typically you will

see microfarads (F) and picofarads (pF) The capacitance of a device depends on the

geometric arrangement of the conductors

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Parallel Plate Capacitor Each plate is connected

to a terminal of the battery

If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires

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Capacitance – Parallel Plates

The charge density on the plates is = Q/A A is the area of each plate, which are equal Q is the charge on each plate, equal with

opposite signs The electric field is uniform between the

plates and zero elsewhere

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Parallel Plate Assumptions

The assumption that the electric field is uniform is valid in the central region, but not at the ends of the plates

If the separation between the plates is small compared with the length of the plates, the effect of the non-uniform field can be ignored

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Capacitance – Parallel Plates, cont. The capacitance is proportional to the

area of its plates and inversely proportional to the plate separation

/o

o

AQ Q QC

V Ed Qd A d

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A parallel-plate Capacitor connected to a Battery

Consider the circuit to be a system Before the switch is closed, the

energy is stored as chemical energy in the battery

When the switch is closed, the energy is transformed from chemical to electric potential energy

The electric potential energy is related to the separation of the positive and negative charges on the plates

A capacitor can be described as a device that stores energy as well as charge

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Capacitance – Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity

Note, this is independent of the charge and the potential difference

4/ o

e e

Q Q RC R

V k Q R k

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Capacitance of a Cylindrical Capacitor

From Gauss’ Law, the field between the cylinders is

E = 2 ke / r

V = -2 ke ln (b/a) The capacitance

becomes

2 lne

QC

bV k a

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20.8 Circuit Symbols A circuit diagram is a

simplified representation of an actual circuit

Circuit symbols are used to represent the various elements

Lines are used to represent wires

The battery’s positive terminal is indicated by the longer line

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Capacitors in Parallel When capacitors are

first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged

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Capacitors in Parallel, 2 The flow of charges ceases when the voltage

across the capacitors equals that of the battery The capacitors reach their maximum charge

when the flow of charge ceases The total charge is equal to the sum of the

charges on the capacitors Q = Q1 + Q2

The potential difference across the capacitors is the same And each is equal to the voltage of the battery

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Capacitors in Parallel, 3 The capacitors can

be replaced with one capacitor with a capacitance of Ceq

The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors

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Capacitors in Parallel, final

Ceq = C1 + C2 + … The equivalent capacitance of a parallel

combination of capacitors is the algebraic sum of the individual capacitances and is larger than any of the individual capacitances

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Capacitors in Series When a battery is

connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery

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Capacitors in Series, 2 As this negative charge accumulates on

the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge

All of the right plates gain charges of –Q and all the left plates have charges of +Q

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Capacitors inSeries, 3 An equivalent capacitor

can be found that performs the same function as the series combination

The potential differences add up to the battery voltage

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Capacitors in Series, final

The equivalent capacitance of a series combination is always less than any individual capacitor in the combination

1 2

1 2

1 2

1 1 1

eq

Q Q Q

V V V

C C C

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Summary and Hints Be careful with the choice of units

In SI, capacitance is in F, distance is in m and the potential differences in V

Electric fields can be in V/m or N/c When two or more capacitors are connected in

parallel, the potential differences across them are the same The charge on each capacitor is proportional to its

capacitance The capacitors add directly to give the equivalent

capacitance

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Summary and Hints, cont When two or more capacitors are

connected in series, they carry the same charge, but the potential differences across them are not the same The capacitances add as reciprocals and

the equivalent capacitance is always less than the smallest individual capacitor

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Equivalent Capacitance, Example

The 1.0F and 3.0F are in parallel as are the 6.0F and 2.0F

These parallel combinations are in series with the capacitors next to them

The series combinations are in parallel and the final equivalent capacitance can be found

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