Electric Potential Energy & Electric Potential Unit 7

62
Electric Potential Electric Potential Energy Energy & Electric Potential & Electric Potential Unit 7 Unit 7

description

Electric Potential Energy & Electric Potential Unit 7. Recall ‘Work’ from earlier. Work done by a force is given by: W = F d cos( q ) or +W: Force is in direction moved -W: Force is opposite direction moved W=0: Force is 90 o to direction moved - PowerPoint PPT Presentation

Transcript of Electric Potential Energy & Electric Potential Unit 7

Page 1: Electric Potential Energy   & Electric Potential  Unit 7

Electric Potential Energy Electric Potential Energy & Electric Potential & Electric Potential

Unit 7Unit 7

Page 2: Electric Potential Energy   & Electric Potential  Unit 7

Recall ‘Work’ from earlier • Work done by a force is given by:

– W = F d cos() or +W: Force is in direction moved -W: Force is opposite direction moved W=0: Force is 90o to direction moved

• Conservative Forces

U = -Wfield

dxxFW )(

Page 3: Electric Potential Energy   & Electric Potential  Unit 7

Electric Potential Energy

General Points:

1) Potential Energy increases if the particle moves in the direction opposite to the field force.

Work will have to be done by an external agent for this to occur

2) Potential Energy decreases if the particle moves in the same direction as the field force on it

ΔU = -Wfield

Page 4: Electric Potential Energy   & Electric Potential  Unit 7

Electrical Potential Energy

Graphical look at EPEThe potential energy is taken to be zero when the two charges are infinitely separated

Page 5: Electric Potential Energy   & Electric Potential  Unit 7

Potential Energy of a System of Charges

Start by putting first charge in position

Next bring second charge into placeNo work is necessary to do this

Now work is done by the electric field of the first charge. This work goes into the potential energy between these two charges.

Now the third & fourth charge are put into place

Work is done by the electric fields of the two previous charges.

The total EPE is then given by (signs matter)

Page 6: Electric Potential Energy   & Electric Potential  Unit 7

Example A test charge is brought separately to the vicinity of a positive charge Q at pt B

A

qrQ

Charge +q is brought to pt A, a distance r from Q

(a) UA < UB (b) UA = UB (c) UA > UB

I) Compare the potential energy of q to that of Q.

(a) (b) (c)

II) Suppose charge q has mass m and is released from rest from the above position (a distance r from Q). What is its velocity vf as it approaches r = ∞ ?

mr

Qqv f

04

1

mr

Qqv f

02

1

0fv

AB

Page 7: Electric Potential Energy   & Electric Potential  Unit 7

Work Done by Uniform Electric Field

Force on charge is

Work is done on thecharge by field

Page 8: Electric Potential Energy   & Electric Potential  Unit 7

Consider a ball of mass, m, placed at a point in space (height, h, above Earth). It would possess a certain PE per unit mass due to it being in the gravitational field of Earth.

If the ball was replaced by a bowling ball of mass, M, it too, would possess the SAME potential energy per unit mass.

ELECTRIC POTENTIAL

Page 9: Electric Potential Energy   & Electric Potential  Unit 7

The electric potential energy per unit charge for some location in an electrical field is called electric potential.

Similarly, 2 different sized charges at the same distance from the charged sphere (green) will have the same EPE/charge.

Page 10: Electric Potential Energy   & Electric Potential  Unit 7

Electric Potential Electric potential is

defined as the potential energy per unit charge at a point in space

Page 11: Electric Potential Energy   & Electric Potential  Unit 7

Comparison:Electric Potential Energy vs. Electric Potential

• Electric Potential Energy (U) - the energy of a charge at some location.

• Electric Potential (V) - tells what the EPE would be if a charge were located there

Page 12: Electric Potential Energy   & Electric Potential  Unit 7

Voltage and Potential Energy

• Positive charge will naturally move towards lower electrical potential energies, lower voltage.

Page 13: Electric Potential Energy   & Electric Potential  Unit 7

General Points for either positive or negative charges:

Positive potential is taken to be higher by definition due to positive test charge.

Page 14: Electric Potential Energy   & Electric Potential  Unit 7

Relation between Potential and Field

The work done by the electric field force in moving a charge from point a to point b is given by

Page 15: Electric Potential Energy   & Electric Potential  Unit 7

Electric Potential for a Point Charge• The direction of the electric field from a point charge is

always radial. We integrate from distance r (distance from the point charge) along a radial line to infinity:

Page 16: Electric Potential Energy   & Electric Potential  Unit 7

What is the electric potential difference between A and B?

Page 17: Electric Potential Energy   & Electric Potential  Unit 7

Rank (a), (b) and (c) according to the net electric potential V produced at point P by two protons. Greatest first.A: (b), (c), (a)B: all equalC: (c), (b), (a)D: (a) and (c) tie, then (b)

Page 18: Electric Potential Energy   & Electric Potential  Unit 7

Question…

The electric potential at point A is _______ at point B

1) greater than

2) equal to

3) less than

Page 19: Electric Potential Energy   & Electric Potential  Unit 7

1) If a positive charge is moved from point A to point B, its electric potential energy

a) Increases b) decreases c) doesn’t change

E

A

BC

Points A, B, and C lie in a uniform electric field.

2) Compare the potential differences between points A and C and points B and C.

a) VAC > VBC b) VAC = VBC c) VAC < VBC

Page 20: Electric Potential Energy   & Electric Potential  Unit 7

Two Charges

Q=-3.5 mCQ=+7.0mC

• In region II (between the two charges) the electric potential is

I II III

1) always positive

2) positive at some points, negative at others.

3) always negative

Page 21: Electric Potential Energy   & Electric Potential  Unit 7

Potential for two charges

Q=-3.5 mCQ=+7.0mC

A

6 m

4 m

How much work do you have to do to bring a 2 mC charge from far away to point A?

Calculate electric potential at point A due to charges

Page 22: Electric Potential Energy   & Electric Potential  Unit 7

Potential of a solid conducting sphere (radius R) with charge +Q

Find V at the following locations:

+QR

i) At r > R

ii) at r = R

Page 23: Electric Potential Energy   & Electric Potential  Unit 7

iii) at r < R

R

Page 24: Electric Potential Energy   & Electric Potential  Unit 7

E vs V graph for conductor

Page 25: Electric Potential Energy   & Electric Potential  Unit 7

V for uniformly charged, nonconducting sphere (radius R)

R

a) Find ΔV for r > R moving from ∞ to a distance r from center

Page 26: Electric Potential Energy   & Electric Potential  Unit 7

b) Find ΔV moving from ∞ to a distance r where r < R.

R

Page 27: Electric Potential Energy   & Electric Potential  Unit 7

04/22/23

Equipotentials

Page 28: Electric Potential Energy   & Electric Potential  Unit 7

Equipotential (EP) Surfaces & Their Relation to Electric Field

An equipotential surface is a surface on which the electric potential is the same everywhere. The EP surfaces that surround the point charge +q are spherical. The electric force does no work as a charge moves on a path that lies on an EP surface, such as the path ABC. However, work is done by the electric force when a charge moves along the path AD.

Page 29: Electric Potential Energy   & Electric Potential  Unit 7

Equipotential Surfaces & Their Relation to Electric Field

The radially directed electric field of a point charge is perpendicular to the spherical equipotential surfaces that surround the charge. The electric field points in the direction of decreasing potential.

Equipotential surfaces (in blue) of an electric dipole. The surfaces are drawn so that at every point they are perpendicular to the electric field lines (in red) of the dipole.

Page 30: Electric Potential Energy   & Electric Potential  Unit 7

Consider two conducting spheres with differing radii Ra and Rb sitting on insulating stands far apart. The sphere with radius Ra has an electric charge +Q. If we connect a thin, conducting line between the spheres, then disconnect it, what are the charges on the spheres?

Page 31: Electric Potential Energy   & Electric Potential  Unit 7

2 identical spheres question

• +Q +Q/2 higher V?

• Attach wire btw spheres…what happens?

• What is final charge of each?

Page 32: Electric Potential Energy   & Electric Potential  Unit 7

Potential difference between charged plates

Page 33: Electric Potential Energy   & Electric Potential  Unit 7

Capacitance

Units = Farads (F)

When the battery is connected to the pair of plates, charges will flow until the top plate’s potential is the same as the + side of the battery, and the bottom plate’s potential is the same as the – side of the battery. No potential difference.

+ -

Describes how much charge an arrangement of conductors can hold for a given voltage applied.

Q is the amount of charge on a plate and ΔV is the voltage applied to the plates

Page 34: Electric Potential Energy   & Electric Potential  Unit 7

Work to charge conductor

• Consider a spherical uncharged conductor, radius R. After a small amount of charge is placed on conductor, its potential becomes V = kQ/R (where V∞=0).

• To further charge conductor, work must be done to bring additional increments of charge, dQ, to place on surface. W = ΔV dQ…the amount of work increases as each dQ is added and sphere becomes more charged.

Page 35: Electric Potential Energy   & Electric Potential  Unit 7

EPE of conductor

Page 36: Electric Potential Energy   & Electric Potential  Unit 7

Capacitance for Parallel Plates– The E field is constant– The geometry is simple, only

the area and plate separation are important.

• To calculate capacitance, we first need to determine the E-field between the plates. We did this using Gauss’ Law:

Total charge qon inside of plate

E and dA parallel

VV

area A

separationd

Page 37: Electric Potential Energy   & Electric Potential  Unit 7

Need to find potential difference. Since E=constant

Total charge qon inside of plate

E and dA parallel

VV

area A

Page 38: Electric Potential Energy   & Electric Potential  Unit 7

How large is 1 Farad?

d

AC o

If a parallel plate capacitor has plates that are separated by only 1mm, in order to achieve 1F, the area of each plate would be…

A = 1.1x108m2

This corresponds to about 6 miles on each side of plate!

Obviously, this is impractical to achieve large capacitance. Therefore, what do they do?

Page 39: Electric Potential Energy   & Electric Potential  Unit 7

Energy per unit Volume

It is necessary at times to relate energy per unit volume to electric field of capacitor (parallel plate)

Page 40: Electric Potential Energy   & Electric Potential  Unit 7

Example

a) Find charge on capacitora) Find charge on capacitor

b) Find energy stored in capacitor

Charge 8.0uF capacitor (C1) by connecting it to a 120V potential difference.

Now remove the power supply.

Page 41: Electric Potential Energy   & Electric Potential  Unit 7

c) Now connect C1 to another capacitor, C2= 4.0uF, initially uncharged.

What will be the potential difference across each capacitor & charge on each after equilibrium is reached?

++++++ ++++++

- - - - - - - - - -

C1 C2

Conservation of charge

Page 42: Electric Potential Energy   & Electric Potential  Unit 7

Method for finding C for various geometries of plates

1) We are trying to calculate C where C = Q / ΔV2) In order to acquire ΔV, we must use

b

a

r

r

EdrV

3) Therefore we must find expression for E first using Gauss’ Law.

4) Find E, then ΔV, and then C.

Page 43: Electric Potential Energy   & Electric Potential  Unit 7

Find capacitance of concentric cylindrical conductors with radius a (inside) & radius b.

Inside charge is +, outside is -

Field is radially outward

Page 44: Electric Potential Energy   & Electric Potential  Unit 7

End view

Page 45: Electric Potential Energy   & Electric Potential  Unit 7
Page 46: Electric Potential Energy   & Electric Potential  Unit 7

+Q-Q

b

a

Spherical Capacitor (2 concentric spheres, inner radius a &

outer radius b as shown)

Page 47: Electric Potential Energy   & Electric Potential  Unit 7
Page 48: Electric Potential Energy   & Electric Potential  Unit 7

Combination of Capacitors&

Equivalent Capacitance (CEQ)

1) Capacitors in Parallel & CEQ

Page 49: Electric Potential Energy   & Electric Potential  Unit 7

All top plates are at same potential and so are bottom plates, so…

C C C

V

C1 C2 C3

Consider 3 identical capacitors in parallel connected to battery of voltage, V. Find CEQ

Page 50: Electric Potential Energy   & Electric Potential  Unit 7

Splitting capacitor into 3 separate capacitors in parallel all with equal potential difference btwn them (same as battery)

Page 51: Electric Potential Energy   & Electric Potential  Unit 7

Capacitors in Series

Charge on each capacitor is the same, Q.

If you place 2 capacitors in series, the charge remains the same, but the potential difference is less for each capacitor

Page 52: Electric Potential Energy   & Electric Potential  Unit 7

• Consider the individual voltages across each capacitor Since q

is the same for each

The sum of these voltages is the total voltage of the battery, V

Page 53: Electric Potential Energy   & Electric Potential  Unit 7

Combo Example

Find Ceq, Q1, Q2, Q3, V1, V2, V3, & Qtotal

A 12 battery is connected to the combination of A 12 battery is connected to the combination of capacitors as shown.capacitors as shown.

12V

8uF

2uF 4uF

C1

C2 C3

Page 54: Electric Potential Energy   & Electric Potential  Unit 7

DIELECTRICSDIELECTRICS

• If the strength of the electric field between the plates of an air filled capacitor becomes too strong, then the air can no longer insulate the charges from sparking (discharging) between the plates. For air, this breakdown occurs when the electric field is greater than 3x106 V/m. (this is what occurs during a lightning strike)…V/m is equivalent to N/C.

In order to keep this from happening, an insulator, or dielectric, is often inserted between the plates to reduce the strength of the electric field, which yields a larger capacitance.

E

Page 55: Electric Potential Energy   & Electric Potential  Unit 7

Why does dielectric reduce E?Dielectric material is polar and molecules polarize as shown.

Electrical forces create a torque to rotate and align molecules

The charge alignment creates an E-field within the material which OPPOSES the original E-field between the plates.

Page 56: Electric Potential Energy   & Electric Potential  Unit 7

The dielectric is measured in terms of a dimensionless constant, κ (greek kappa) ≥ 1. (see table)

Page 57: Electric Potential Energy   & Electric Potential  Unit 7

Assuming a capacitor is charged with no power source present, dielectric reduces E which reduces V (according to V = Ed) while d remains constant. If V reduces, then C increases (according to C = Q / V)

Page 58: Electric Potential Energy   & Electric Potential  Unit 7

exampleA parallel-plate air capacitor is charged by placing a 90-V battery across it. The battery is then removed. An insulating, dielectric fluid is inserted between the plates. The voltage across the capacitor is now 28V. What is the dielectric constant of the fluid?

Page 59: Electric Potential Energy   & Electric Potential  Unit 7

Example 2A parallel-plate air capacitor holds a charge of 30nC when a voltage of V is placed across its plates. If the battery is not removed and a dielectric fluid is inserted between the plates, the charge on the plates increases to 87nC. Find the dielectric value.

Page 60: Electric Potential Energy   & Electric Potential  Unit 7

The charged plates of an air-filled capacitor are 10 cm by 20 cm and the gap between the plates is 6 mm. What is the capacitance when its gap is only half-filled with a dielectric having κ = 3.0?

We treat this problem as 2 dielectrics in SERIES

Page 61: Electric Potential Energy   & Electric Potential  Unit 7

Dielectric application - computer key on keyboard

When the key is pressed, the plate separation is decreased and the capacitance increases. Each key corresponds to a different capacitance.

Page 62: Electric Potential Energy   & Electric Potential  Unit 7

Dielectric application: Stud-finder(how you doin!)

The dielectric constant of wood (and of all other insulating materials, for that matter) is greater than 1; therefore, the capacitance increases. This increase is sensed by the stud-finder's special circuitry, which causes an indicator on the device to light up.