Idea Transcript
CHAPTER 23 ELECTRIC POTENTIAL •
Potential difference and electric field
•
Potential difference between two parallel plates
•
Potential due to a single point charge EFM08AN1.MOV
•
Potential due to a collection of charges • Work done bringing charges together
•
Potential for continuous charge distributions • Charged, hollow sphere • Uniformly charged ring
When a system, e.g., you, a compressed spring, an
•
Equipotential surfaces
the same as the loss of potential energy:
•
Electrostatic potential energy
Work done and potential energy.
electric field, does (positive) work, it loses potential energy and the amount of work done by the system is δW = −δU.
Work done and potential energy ... a ! δℓ b ! ! F = mg ! g
! δℓ ! + ! F = qE
! δℓ
+
! E
- - - - - - - - - Let’s look at the similarity between electric and gravitational fields. The work done by the g-field in moving the mass from a → b is: b! ! δW = ∫ F • dℓ = mgδℓ ( > 0), i.e., positive work.
+
q#
a
b ! F
gravitational field in moving from a → b is:
Note: the work done by the g-field ( δW) in moving the mass from a → b is the same as the work you do in raising the mass from b → a.
charge from a → b is ! ! δW = F • δℓ = −δU,
potential energy of the charge in the field is: ! ! ! ! δU = (Ub − Ua ) = − F • δ ℓ = −q #E • δ ℓ . We define the electric potential (V) as the potential energy per unit charge, i.e.,
V=
U . q#
So the potential difference between b and a
δU = Ub − Ua = −mgδℓ ( < 0), i.e., a loss. ∴δW = −δU = −(Ub − Ua ).
done by the E-field in moving the
where δU (= Ub − Ua ) is the change in potential energy ! ! ! of the charge in the E-field. But F = q #E , so the change in
a
The change in potential energy of the mass in the
If a (positive) charge moves through a ! ! displacement δℓ in a field E, the work
is given by:
δV = Vb − Va = Vba ! ! δU δV = = − E • δℓ . q#
Since δV < 0, Va > Vb , i.e., the charge moves from a higher to a lower potential.
We will work problems in Cartesian (x,y,z) and polar (r, θ,φ) coordinate systems. [1] Cartesian coordinates:
UNITS:
! ! E ⇒ (E x ,E y ,E z ) and δℓ ⇒ (δx, δy, δz) ! ! δV = − E • δℓ = −(E ˆi + E ˆj + E kˆ ) • (δxˆi + δyˆj + δzkˆ ) x
i.e.,
z
δV = −(E x .δx + E y .δy + E z .δz).
∴E x = − i.e.,
y
∂V ∂V ∂V , Ey = − , Ez = − ∂x ∂y ∂z
! ⎛ ∂V ∂V ˆ ∂V ˆ ⎞ E = − ⎜ ˆi + j+ k⎟ . ∂y ∂z ⎠ ⎝ ∂x
These are the basic relationships between the electric ! field, E, and the electric potential, V, in Cartesian coordinates.
If E ⇒ N/C and
r ⇒ m
then: V ⇒ volts (V). But, by definition: Ex = −
dV , etc., dx
then: E ⇒ V/m, which means that N/C ≡ V/m. (It is more usual to use V/m as the unit of electric field.)
Conventional definition of work done in an electric field ... DISCUSSION PROBLEM:
" δℓ +
q! ~10,000V
a
b " F
120V
The work done by the field in moving the charge from a → b is δW = −δU = −(Ub − Ua )
= −(q ! Vb − q ! Va ) = q ! (Va − Vb ). (Remember, by definition ⇒ V = U q ) !
Conventionally, when a charge moves from a → b we
When you charge a balloon by friction, its electric
write the work done by the field as:
potential is ~10,000V, but it is safe to handle! And yet,
δW = q ! (Va − Vb ) = q ! Vab , where Vab is the potential difference between the start
a typical socket operates at a potential of 120V but will give you a (potentially!) fatal shock. *
What’s the difference?
*
Why is the socket more “shocking”?
point (a) and the end point (b). (Note also if the charge was released and free to move in the field, δK = δW.) Therefore, the work done by you in moving a charge from a → b is: δW = −q ! Vab = −q ! (Va − Vb ).
+σ
Example using Cartesian coordinates ... • Potential between two parallel charges plates ! E = E x ˆi 1 2 2 1 − ΔV + b d ! ℓ a d ˆj
V1
! between a and b (displacement ℓ ) ! in a field E = E ˆi produced x
between two parallel, infinitely large charged plates, spaced a distance d apart. Along the displacement, the change in potential is: ! ! dV = − E • d ℓ = −E xˆi • (dxˆi + dyˆj) = −E xdx. b
b
a
a
∴Vb − Va = ∫ dV = −E x ∫ dx = −E x x = − σ ∴Vb = Va − x ε#
σ x ε#
" E = E x ˆi
2
V2
d V d
ΔV
σ x ε! For the pair of plates Vb = Va −
(V2 − V1 ) = ΔV = −
V1
Find the potential difference
! ℓ = xˆi + yˆj ˆi
−σ
1
i.e., V1 > V2 .
σ d, ε!
x
V2 • ΔV is independent of y, it depends only on σ and d. Thus, ΔV is the same between any point on plate 1 and any point on plate 2 . This means that the potential is constant over an infinitely charged plate. •
The work done by the field in moving a charge q
from a → b is: δW = q(V1 − V2 ) > 0, so a +ve charge moves from a position of higher potential ( V1) to lower potential ( V2).
V a
Question 23.1: Shown here is a plot of electric potential versus distance in a region where there is an electric field.
b
Where is the magnitude of the electric field greatest?
c
V a b
c
x e d
By definition: Ex = −
dV , dx
so E x is a maximum at d and e.
x e d
Question 23.2: The facing surfaces of two large parallel
+σ
conducting plates separated by 10.0 cm have uniform surface charge densities +σ and −σ. The difference in potential between the plates is 500V. (a) Which plate has the higher potential? (b) What is the magnitude of the electric field between the plates? (c) An electron is released from rest next to the
(a,b) Since the field moves a +ve
−σ
charge from the +σ plate to the −σ 1
e−
2
plate, the +σ plate is at the higher potential.
10 cm
ΔV = (V1 − V2 ) = 500V = −E.d, ∴ E = ΔV d = 500 0.1 = 5000 V/m.
(c) Work done by the field is W2→1 = qV21. But V21 = (V2 − V1 ) = −500 V. ∴W2→1 = −1.6 × 10−19 C × (−500 V) = 8.0 ×10 −17 J. (d) The change in potential energy of the electron:
negatively charged plate. What is the work done by the
ΔU = U1 − U2 = qV1 − qV2 = q(V1 − V2 )
electric field as the electron moves from the negatively
= −1.6 ×10 −19 C × 500 V = −8.0 × 10 −17 J.
charged plate to the positively charged plate? (d) What is the change in potential energy of the electron as it moves from one plate to the other? (e) What is its kinetic energy when it reaches the positively charged plate?
(e) Mechanical energy is conserved: ∴ΔK = −ΔU = 8.0 × 10−17 J. 0 1 2ΔK 2 2 ΔK = m(v − v ! ). ∴v = = 1.33 × 107 m/s. 2 m Note also, from the work - kinetic energy theorem: ΔK = W2→1 = qV21.
From earlier, the components of the electric field are related to the electric potential in the (x, y, z) coordinate system by the relationships: ∂V ∂V Ex = − , Ey = − , ∂x ∂y
Ez = −
Given V = (4xz − 5y + 3z 2 ), we have Question 23.3: The electric field in a certain region of 3-dimensional space varies as: V = (4xz − 5y + 3z 2 ) volts. What is the electric field at (2, −1,3), where all distances are in meters.
Ex = −
∂V = −4z, ∂x
Ey = −
∂V = 5, ∂y
Ez = −
∂V = −4x − 6z. ∂z
! So, at r = (x, y,z) ⇒ (2, −1,3) m,
E x = −4z = −12 V/m, E y = 5 V/m, E z = −4x − 6z = −26 V/m. ! ∴ E = −12ˆi + 5ˆj − 26 kˆ V/m.
(
)
∂V . ∂z
Example ... electric potential for a point charge:
[2] Polar coordinates If the electric field has radial symmetry, i.e., it depends ! only on r , e.g., a point charge, then ! Q E (r) ⇒ E r ˆr = k 2 ˆr . ˆr r ! ! d r + For a radial displacement dr (in
! But dr // ˆr ,
the ˆr direction): ! ! ! dV(r) = − E(r) • d r = −E r ˆr • d r . =1 ! ! ∴ ˆr • d r = ˆr d r cos0 = dr.
Therefore, potential difference between radii r2 and r1 is r2
V21 = V(r2 ) − V(r1 ) = − ∫ E r dr. r1
But dV(r) = −E r dr, so the radial electric field is dV(r) Er = − . dr Again, we have simple relations between the electric field E r and the electric potential V(r).
The electric field of a point Q
+ a• d!r • b
ˆr
charge is: ! Q E(r) = k 2 ˆr . r
! For a small displacement dr in the radial direction ( ˆr ), the change in potential is: ! Q Q ! ! dV(r) = − E(r) • d r = −k 2 ˆr • d r = −k 2 dr. r r Q Q ∴V(r) = −k ∫ 2 dr = k + V" , r r where V" is an integration constant. If we define the electric potential at infinity as zero, i.e., V(r → ∞) = 0, then V" = 0. So, the absolute electric potential at r is Q V(r) = k , r ⎡1 1⎤ ∴Vb − Va = kQ ⎢ − ⎥ (i.e., < 0 if Q is +ve) ⎣ rb ra ⎦
Go from a → b by different routes. The potential at any point a distance r from a
“equipotentials”
a
Electric potential (V)
point charge is: Q V(r) = k . r
x b
Since ra = rx , Va = Vx , so the potential difference Vab = Vxb ,
y
i.e., the potential difference between two points does not depend on the path between them only the potentials at the end points. The work done by you in moving a
x V= k
Q r
charge q from a → b by the two different routes is:
The electric potential for a positive charge. If the charge
[1] Wa →b = −q(Va − Vb ) = −qVab . [2] Wa →x→b = [ −q(Va − Vx )] + [ −q(Vx − Vb )]
is negative, the potential looks like a “hole” rather than a
= −q(Vx − Vb ) = −qVxb .
“hill”. Note that as x (and y) → ±∞, V → 0.
But Vab = Vxb .
∴Wa → b = Wa →x→ b .
So, the work done by you ( = −qΔV) in moving a charge from one point to another does not depend on the path ... only on ΔV.
The electric force is a conservative force. Here are two reasons ... [1] The work done by the electric force in moving a charge from one point to another is independent of the path ... a property of a conservative force. [2] We can write a potential (energy) function, which Question 23.4: Is the electric force a conservative or non-conservative force?
can only be done for conservative forces. Q For a point charge V(r) = k . Therefore, the potential r energy of a test charge q ! in the field due to a point charge Q is qQ U(r) = q ! V(r) = k ! . r i.e., a simple function of r. No matter what the source, the electric force is a conservative force.
+
Question 23.5:
r2 +
2
+
+
2
r1 +
+
1
A proton is fired towards a helium nucleus. If the speed of the proton at point 1 is v1 and its speed at point 2 is v 2, which of the following statements is correct? A: v 2 < v1. B: v 2 > v1. C: v 2 = v1.
1
Use the conservation of mechanical energy: K2 + U2 = K1 + U1. The potential energy of the proton (with charge q) in the field of the helium nucleus at 1 is U1 = qV1, and its potential energy at 2 is U2 = qV2 . Since V = k
Q and r2 < r1, then V2 > V1. r
∴U2 > U1. So, K2 < K1, i.e.,
v 2 < v1,
which means A is the correct choice.
A
+2µC 3m
3m 3m B +2µC
Question 23.6: Points A, B and C are at the vertices of
V= k C
Q r
Point charges of +2.00 µC are fixed at A and B.
(a) Potential at C is due to both QA and QB ( = Q) Q Q Q VC = k A + k B = 2k , r r r
(a) What is the electric potential at point C?
where r is the length of the sides. Remember V is a
an equilateral triangle whose sides are 3.00 m long.
scalar. (b) How much work is required to move a +5.00 µC point charge from infinity to the point C? (c) How much additional work is required to move the +5.00 µC charge from C to the midpoint of side AB?
∴VC
2 × 10 −6 ) ( = 2 × (9 × 10 ) × = 1.2 × 10 4 volts. 9
3
(b) Work done in bringing a charge q ! from ∞ to the point C is: W = −q ! ΔV = −q ! (V∞ − VC ) = q ! VC
(
) (
= 5 ×10 −6 × 1.2 ×10 4 = 6.0 × 10−2 J.
)
A
Potential due to a spherical shell of charge • on a hollow or solid conducting sphere ...
+2µC
+ +R + + + + + +
D× B +2µC
C
(c) The extra work done in moving the charge from C to D is δWC→D = −q ! ( VC − VD ) = q ! (VD − VC ) Q Q Q But VD = k A + k B = 4k r r r 2 2 Q From earlier, VC = 2k . r ⎛ Q Q⎞ Q ∴δWC→D = q ! ⎜ 4k − 2k ⎟ = 2q ! k ⎝ r r⎠ r
( )
(
= 2 × 5 × 10
−6
( )
2 × 10 −6 ) ( ) × (9 × 10 ) × 9
3
Q σ Es = k 2 = ε! R
Q 4πR 2
E(r)
E=0 From earlier:
σ⇒
E=−
dV so dr
Q E=k 2 r r dV = −E.dr,
Q Q ∴V(r > R) = − ∫ E.dr = −k ∫ 2 dr = k r r r >R V(r) Q R
Vs = k
V= k
= 6.0 × 10−2 J.
Q r r
But what about inside the sphere?
TWO POINTS:
Potential due to a spherical shell of charge • on a hollow or solid conducting sphere ... Q σ Es = k 2 = ε! R
[1]
E(r)
E=0
[2]
Q E=k 2 r r
V is constant inside a conducting sphere (i.e., the same as at the surface). Q Q At the surface: Vs = k and Es = k 2 . R R V ∴Vs = Es .R or Es = s . R
As the charge Q on the sphere increases, so do ⎛ ⎞ Vs = k Q R and Es ⎜ = k Q 2 ⎟ . ⎝ R ⎠
Inside the sphere, i.e., for r < R, dV = −E.dr = 0
(
∴V(r < R) = constant.
)
If V is constant inside sphere, no work is done in
Under “normal conditions” the maximum electric field
moving a charge anywhere inside the sphere. Then
obtainable in air before breakdown is
W = −q !ΔV = 0, i.e., ΔV = 0. Q ∴V(r < R) = k ⇒ constant. R
Emax ~ 3 × 106 V/m. This sets a maximum potential and a maximum charge for a spherical conductor (radius R):
V(r) Vs = k
Q R
i.e.,
Vmax = E max .R ~ 3 × 106 R volts.
Q 3 × 106 2 Since, E = k 2 , then Qmax ~ R Coulombs. k R
Q V= k r r
Larger R means larger Vmax and Qmax before breakdown.
Assume the Earth is a sphere. The potential at the surface of a sphere is: Vs = Es .R. For the Earth: Vs = Es .R = (200 V/m ) × (6400 × 10 3 m) Question 23.7: We know that the electric field near the
= 1.28 × 109 volts!
Earth’s surface is ~200 V/m. If the Earth has a radius of about 6400 km, what is the Earth’s electric potential?
Note: in question 22.4, we found that the net charge on the Earth is negative ... −9.11 × 105 C ... so the electric potential at the surface is Vs = k
Q R
σ1. R2
σ σ Also, E1s = 1 and E2s = 2 . ∴E2s > E1s . ε! ε! Therefore, the charge density and the surface electric field are greater at “points”.
R2
R3
R3
R1 1
2
3
We know that V = EsR. Therefore, the maximum 1
2
3
Question 23.8: These objects are charged to their maximum potential (voltage) before breakdown of the
potential (when breakdown occurs) Vmax ∝ R. Since R 3 > R1 > R 2 , the order from largest maximum potential to smallest is: 3 :1: 2
surrounding air. Rank them in order from the one with the largest voltage to the smallest voltage.
Note that at the sharp end of #2 the electric field is greatest; that’s where breakdown first occurs.
Take two charged spheres A ( VA) and B ( VB) with VA > VB ... connect them together by a conducting wire. Q VA = k A RA
q
A
+
B
The force on charge q is: F = qE = −q so
Q VB = k B RB
dV , but VA > VB dℓ
dV < 0, therefore, +ve charges move from A to B, i.e., dℓ
from high potential ( VA) to low potential ( VB).
Question 23.9: A spherical conductor of radius R 1 = 24.0 cm is charged to 20.0 kV. When it is connected by a long conducting wire to a second conducting sphere a great distance away, its potential
V
∴QA and VA decrease
uncharged, what is the radius, R 2, of the second sphere?
+
∴QB and VB increase As charges move from A to B, VA decreases and VB increases. When VA = VB, charges stop moving because when ΔV = 0 then F = 0.
drops to 12.0 kV. If the second sphere was initially
V1 = 20kV V ′ = 12kV 1
R1
R2
1
V2 = 0 V ′ = 12kV 2
2
After the spheres are connected we have: q q V1′ = k 1 = V2′ = k 2 = 12 × 10 3 V, R1 R2
q1
R R i.e., q 1 = (12 × 10 3 ) 1 and q 2 = (12 × 10 3 ) 2 k k
... [1]
The total charge (q1 + q 2 ) is conserved, before they are connected we have (q + q 2 ) V1 = k 1 = 20 × 103 V, R1
(
i.e., q 1 + q 2 = 20 × 10 3
) Rk1 .
Using q1 and q 2 from [1], R R R (12 × 103 ) 1 + (12 × 103 ) 2 = (20 × 103 ) 1 , k k k i.e., (12 × 10 3 )R 2 = (8 × 10 3 )R 1. 8 2 ∴R 2 = R 1 = × 24.0 cm = 16.0 cm. 12 3
q2 R2
R1 R 2 = 2R1
Question 23.10: Initially, two, well separated conducting spheres are given the same charge (q). They are then connected to each other with a conducting wire. If R 2 = 2R1, what are the final charges, q1 and q 2, and charge densities, σ1 and σ 2 , on each sphere?
Electrostatic potential due to a uniformly charged ring:
q2
q1
dQ R2
R1 R 2 = 2R1
If they have the same initial charge, the potential of #1 ( V1) will be greater than the potential of #2 ( V2), because R 1 < R 2 . After connecting the two spheres together,
+ +
r = x2 + a 2
a
+
+ +
x
P × dV
+
charges will flow from #1 to #2 until the potentials are
The potential at the point P due to the element of charge
equal, i.e., V1 = V2 . Then q q q q k 1 = k 2 , i.e., 1 = 2 . R1 R2 R1 2R1
dQ a distance r from P is:
(
∴q 2 = 2q1.
)
The surface area 4πR 2 of #2 is four times the surface area of #1, i.e., A 2 = 4A1. q 2q 1 ∴σ 2 = 2 = 1 = σ1. A 2 4A1 2
dV = k
dQ . r
∴V(x) = ∫ k
dQ . r
But r = x 2 + a 2 ⇒ constant. ∴V(x) =
k x2 + a
dQ = k 2∫
where Q is the total charge on the ring.
Q x2 + a 2
Electrostatic potential due to a uniformly charged ring: dQ
+ +
r = x2 + a 2
a
+
+ +
x
a x
P ×
+
V(x) = k V= k
P × dV
Q a
Question 23.11: What difference would it make, if any,
Q
if the charge Q were NOT uniformly nor symmetrically
x2 + a 2
distributed around the ring? V= k
Q x
A: The potential at any point P on the axis would be larger. x
-2
0
2
4
6
B: The potential at any point P on the axis would be a
Q dV • When x = 0, V(x) = k . Also, E x = − = 0. a dx Q • When x >> a, V(x) ⇒ k , x i.e., the ring looks like a point charge.
smaller. C: The potential depends on the actual distribution. D: It would make no difference.
dQ ⇒ Q r a
P × V
“equipotentials”
V
Take an extreme scenario, i.e., with all the charge concentrated at one point on the ring. The total potential at P due to dQ is V= k
dQ Q ⇒k , r r
x 0 0
y 0
i.e., the same as before! Since potential is a scalar quantity it makes no difference if the charge is concentrated in one region or distributed around the ring; it only depends on r, the distance of P
Three dimensional plot of the electric potential for two
from the ring.
equal and opposite charges.
Therefore, the answer is D.
y
X
Y
Question 23.12: Several equipotentials are shown labelled in volts. The spacing of the grid is 1.00cm. What is the magnitude and direction of the electric field at
x
20V
(a) X,
15V
(b) Y?
10V
y
5V
1 cm
0V
X
1 cm
Y
x
20V 15V 10V 5V
1 cm
0V
1 cm
The electric fields at X and Y are parallel to the x-axis. ! ∂V ˆ ∴E = − i. ∂x ! ΔV (15 V − 5 V) ˆ 10 V ˆ (a) at X: E = − ! = − i=− i Δx 0.01 m 0.01 m = −1000 ˆi V/m. The negative sign means the electric field is in the −x ! direction. (Note also that E is always directed from higher potential to lower potential ... it’s the direction a +ve charge would move.)
y
X
Y +4.2µC x
20V 15V
−4.2µC
10V 5V
1 cm
0V
1 cm
(b) at Y:
! ΔV (5V − 15V) ˆ 10V ˆ E=− ! =− i= i Δx 0.02m 0.02m = +500 ˆi V/m.
The positive sign means the electric field is in the +x direction.
2.50 m
2.50 m
2.50 m −4.2µC
Question 23.12: Point charges, q1 = q 3 = −4.20 µC and q 2 = +4.20 µC are fixed at the vertices of an equilateral triangle, whose sides are 2.50 m long. What is the electrostatic potential energy of this ensemble of charges?
B 2.50 m
B 2.50 m
2.50 m
A 2.50 m q 1 = −4.2µC
q 2 = +4.2µC
A q 1 = −4.2µC
q 2 = +4.2µC C
q 3 = −4.2µC
Now bring in a charge q 3 to the point C. The work done in bringing charge q 3 from ∞ to C is: W3 = −q 3(V∞ − VC ),
The potential energy of an ensemble of charges is simply equal to the work done in bringing the charges together.
where VC is the potential at C due to the charges at A
Assume charge q1 is at position A and bring in charge q 2
and B.
to B from ∞. The potential at B due to charge q1 at A is: q VB = k 1 . rAB So, the work done in bringing a charge q 2 from ∞ to B is: W2 = −q 2ΔV = −q 2 (V∞ − VB ) = −q 2 (0 − VB ) qq = k 1 2 = (9 × 109 ) r12
(−4.2 × 10−6 )(4.2 × 10 −6 )
= −6.35 ×10 −2 J.
⎛ q q ⎞ qq q q ∴W3 = −q 3⎜ 0 − (k 1 + k 2 )⎟ = k 1 3 + k 2 3 ⎝ r13 r23 ⎠ r13 r23 = (9 × 109 )
(−4.2 × 10 −6 )(−4.2 ×10 −6 ) 2.50 +(9 × 109 )
2.50
(4.2 × 10 −6 )(−4.2 ×10 −6 ) 2.50 = 0.
So, the total work done = W2 + W3 = −6.35 × 10 −2 J. ?
Electrostatic potential energy of a charged sphere The electrostatic potential energy of a charged conducting sphere is equal to the amount of work we do in putting the dq
q
R
+
charge onto the sphere. If the sphere already has a charge q, the work done in bringing a charge dq from ∞
onto the sphere is dW = −dq(V∞ − V), where V is the potential of the sphere. But V∞ = 0 and V = k q R , ∴dW = k
( R)dq.
( )
q
The total work done to charge the sphere from 0 → Q is: Q
( )
Q
k ⎡q 2 ⎤ 1 Q2 1 W = ∫ k q R dq = = k R = QV, 2 ⎢ ⎥ R⎣ 2 ⎦0 2 0 Q where V ( = k R ) is the potential of the fully charged sphere. 1 ∴U = W = QV. 2
Question 23.13: An isolated spherical conductor with a radius of 10.0 cm is charged to 2.00 kV. (a) What is the electrostatic potential energy of the sphere? (b) What is the charge on the surface of the sphere?
Q
(a) We’ve just shown that the electrostatic potential energy of a 1 charged sphere is: U = QV. 2 Q VR But, V = k , i.e., Q = . R k R
Substituting for Q, we find 1 VR ⎞ V2R U= ⎛ V= 2⎝ k ⎠ 2k (2.00 × 103 )2 × 0.10 = = 2.22 × 10 −5 J. 9 2 × 9 ×10 This is the amount of work we do in charging the sphere to a potential of 2.00 kV. (b) The charge on the sphere is (from above): Q=
RV 0.10 × 2.00 × 103 = = 2.22 × 10−8 C 9 k 9 × 10 = 22.2 nC.