electricity phy1013s potential gregor leigh [email protected]

ELECTRICITY

PHY1013S

POTENTIA

L

Gregor [email protected]

ELECTRICITY ELECTRIC POTENTIALPHY1013S

2

ELECTRIC POTENTIAL

Learning outcomes:At the end of this chapter you should be able to…

Distinguish carefully between electrical potential energy, potential difference and potential (and other terminology).

Determine the electric potentials at various points in fields due to specific charge distributions, and illustrate these potentials using several graphical representations.

Calculate electric potential from electric field & vice

versa.

Apply the law of conservation of energy to determine the behaviour of charged particles in electric fields.


3

GRAVITATIONAL POTENTIAL DIFFERENCE

Objects have different potential energies at different points (heights) in a gravitational field.

2 kg 4 kg 80 g0.5 m

1 m


4


The actual difference in potential energy between the two points depends on the mass being moved.

U = 2 1 9.8 J U = 0.08 1 9.8 J

0.5 m

1 m

2 kg

4 kg

80 g

U = 4 0.5 9.8 J


5


But if instead we consider the difference in the potential energy per unit of mass (i.e. for each kilogram) between the two points, we are considering a property of the field.

0.5 m

1 mU = 9.8 J per 1 kg

2 kg

4 kg

80 g

U = 4.9 J per 1 kg


6

U = 9.8 J per 1 kg

U = 4.9 J per 1 kg


We might call this difference in gravitational potential energy per unit of mass the gravitational potential difference between the two points:

0.5 m

1 mG = 9.8 J/kg

G = 4.9 J/kg

UGm


7

Units: [J/kg]

And hence we might (in the interests of obfuscation) talk about the “greggage” between two points in a gravitational field.


0.5 m

1 mG = 9.8 G

G = 4.9 G

UGm

G = 9.8 J/kg

G = 4.9 J/kg

= [greg, G]


8

ELECTRIC POTENTIAL ENERGY

Electric potential energyElectrostatic potential energy

U

But only changes or differences in potential energy are meaningful. As a field does work on a charged particle the particle loses potential energy:

The energy is the energy of a system of charges,but you will hear “the energy of a particle…”.

The work done is done by the force on the particle due to the other charge(s),but you will hear “the work done by the field…”.

Notes:

U = Uf – Ui = –W

elec


9

ELECTRIC POTENTIAL DIFFERENCE

The difference in the amount of electric potential energy per unit of charge between one point and another in an electric field is known as…

difference potential

Electric potential differencePotential difference

WVq

Units: [J/C = volt, V]

Hence electric potential difference is sometimes (colloquially) referred to as the voltage between two

points, or “across” a component in a circuit.

UVq

…the potential difference between those points.


10

ELECTRIC POTENTIAL, V

Using infinity as our reference (zero) point, Ui = 0,

Electric potential (at a point)Potential

(??!)UVq

U = Uf – Ui = Uf – 0 = Uf = –W

Hence the potential at a point is given by: WV

q

But what could possibly be meant by “the potential at one point”?

or simply: U = –W

and hence:

ELECTRICITY

but sometimes it is easier to use the electron volt (eV): 1 eV = 1.6 10–19 J.

ELECTRIC POTENTIALPHY1013S

11

ELECTRIC POTENTIAL, V

Notes: Electric potential is a property of the field (or, more specifically, it depends on the source charges and their geometry). Though we use a “probe” charge to measure it, like the field itself, potential exists whether an “intruder” charge is there to experience it or not.

Electric potential is a scalar quantity.

Like potential difference, potential is measured in volts (joules/coulomb).

w = qV [J = C V]…


12

V1 = 80 V

EQUIPOTENTIAL SURFACES

An equipotential surface is a collection of points which all have the same potential.

No net work is done by or against the electric field when a charge movesbetween two points on the same equipotential surface (whatever route it follows).

The work done moving a charge from one equipotential surface to another is independent of the path taken.

Equipotential surfaces lie perpendicular to field lines.

c

ba

V2 = 60 V

V3 = 40 V


13

EQUIPOTENTIAL CONTOURS

–

+

+

equipotential contours

Equipotential surfaces (which lie perpendicular to the field lines) can also be represented as equipotential contours:


14

CALCULATING THE POTENTIAL FROM THE ELECTRIC FIELD

A particle with charge q moves from initial position i to final position f along an arbitrary path in a non-uniform field…At any point the force acting on the particle is F qE

and the differential work done by the field during a displacement is dW F ds qE ds

Integrating over the whole path for the net work done by the field, we get

f

ifi

W q E ds

i

f

+

q

F

ds

+


15

F qE

dW F ds qE ds

f

ifi

W q E ds

And since (from our definition of V)if

f i

WV V

q

If i is at infinity (where Vi = 0), then the

potential at any point relative to infinity is f

iV E ds

CALCULATING THE POTENTIAL FROM THE ELECTRIC FIELD

+

qds

F

i

f


16

POTENTIAL DUE TO A POINT CHARGE

Letting a test charge move radially inwards from to P…

r

V E dr'

20

14

rqV dr'

r'

0

14

rqV

r'

0

14

qV

rand hence…q

q0

P

r'r

+

dsdr

E

cos180E ds E ds E ds

'E dr


17

POTENTIAL DUE TO MULTIPLE POINT CHARGES

Hence:

ii

V V

0

14

i

i i

qV

r

The sign of each qi determines the sign of its Vi,

but the addition is algebraic, not vector!

For a continuous charge distribution:

Notes:

0

14

dqV

r

The principle of superposition applies, i.e. .


18

P

POTENTIAL DUE TO AN ELECTRIC DIPOLE

When r >> s…

0 04 4q q

Vr r

+

–q

+q

z

O

s

+

–

The potential at P is…

0

1 14

qV

r r

04r rq

Vr r

r+

r–

r


19

POTENTIAL DUE TO AN ELECTRIC DIPOLE

When r >> s…

+

–q

+q

z

r+

r–

r

s

+

04r rq

Vr r

r– – r+ s cos

r–r+ r2

20

cos4

q sVr

20

cos14

pV

r

Note: V = 0 for all points in the plane defined by = 90°

r– – r+ s cosqs p

–

P

O


20

dW F ds

CALCULATING THE ELECTRIC FIELD FROM THE POTENTIAL

A positive test charge moves along the path interval between two equipotential surfaces.

The potential difference between the surfaces is dV. The work done by the field E is dW = –q dV

and also

From which we get cos dVEds

s

E

ds

V

V + dV

qE ds

cosqE ds

q++


21

cos dVEds

CALCULATING THE ELECTRIC FIELD FROM THE POTENTIAL

…

E cos is simply the component of the electric field in the direction of ,

ds

Taking this direction successively along the three principle axes, we derive the components of E:

sVEs

; ;x y zV V VE E Ex y z

And if the electric field is uniform… VEs

so we can write

s

q

E

+ ds

V

V + dV


22

ELECTRICAL POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES

The electric potential energy of a system of fixed charges is equal to the work done by an external agent in assembling the system.

While q2 is still at , the potential at the

position P which will be occupied by q2 is1

0

14

qV

r

Bringing q2 in from to P, requires work: agent 2W W q V

P+

q1to charge q2

r


23

Since and and …

ELECTRICAL POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES

1

0

14

qV

r agent 2W q V

+q2

+q1

r

1 2

0

14

q qU

r

… the potential energy of the pair of charges is thus

If q1 and q2 are unlike charges the work is done by the

field, and the system has negative potential energy.

agentU W


24

Excess charge on an isolated conductor distributes itself on the surface of the conductor in such a way that the field inside the conducting material is zero (regardless of whether the conductor has an internal cavity – which may or may not contain a net charge).

POTENTIAL OF A CHARGED ISOLATED CONDUCTOR

0f

iV E ds

Thus the potential is the same at all points on and inside the conductor.


25

For a charged spherical conductor (solid or hollow) of radius r0 …

POTENTIAL OF A CHARGED ISOLATED CONDUCTOR

rr0

V(r)

E(r)

rr0

1r

21

r

… And remembering that the electric field is the derivative of the potential…

sVEs


26

CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR

The surface of the conductor is an equipotential surface.


27


The further one moves away from a tiny conductor, the more the equipotential surfaces resemble those around a point charge, i.e. they become spherical.


28


In order to “morph” into spheres, equipotential surfaces near small-radius convex surface elements have to be closer than they are near “flatter” parts of the surface.


29


Equipotential surfaces are closest together where the electric field is strongest.


30


Thus on an isolated conductor the concentration of charges and hence the strength of the electric field is greater near sharp points where the curvature is large.


31

For an isolated, uncharged conductor in an external field, the free charges (electrons) distribute themselves on the surface of the conductor in such a way that …

the net field inside the conducting material is zero;

the net electric field at the surface is perpendicular to the surface.

ISOLATED CONDUCTOR IN AN EXTERNAL ELECTRIC FIELD

E = 0

electricity phy1013s potential gregor leigh [email protected]

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