MIT Chapter 4 - MIT Physics Textbook PDF

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Chapter 4 Electric Potential ! 4.1! Potential and Potential Energy ............................................................................. 4-3! 4.2! Electric Potential in a Uniform Field ................................................................... 4-7! 4.3! Electric Potential due to Point Charges ............................................................... 4-8! 4.3.1 Potential Energy in a System of Charges...................................................... 4-10! 4.4! Deriving Electric Field from the Electric Potential ........................................... 4-12! Example 4.4.1: Calculating Electric Field from Electric Potential ........................ 4-13! 4.5! Gradients and Equipotentials ............................................................................. 4-13! 4.5.1: Conductors and Equipotentials .................................................................... 4-16! 4.6! Continuous Symmetric Charge Distributions .................................................... 4-17! Example 4.61: Electric Potential Due to a Spherical Shell .................................... 4-17! Example 4.6.2 Conducting Spheres Connected by a Wire .................................... 4-19! 4.7! Continuous Non-Symmetric Charge Distributions ............................................ 4-20! Example 4.7.1: Uniformly Charged Rod ............................................................... 4-20! Example 4.7.2: Uniformly Charged Ring .............................................................. 4-22! Example 4.7.3: Uniformly Charged Disk .............................................................. 4-23! 4.8! Summary ............................................................................................................ 4-26! 4.9! Problem-Solving Strategy: Calculating Electric Potential ................................. 4-27! 4.10!Solved Problems ................................................................................................ 4-30! 4.10.1 Electric Potential Due to a System of Two Charges ................................... 4-30! 4.10.2 Electric Dipole Potential ............................................................................. 4-31! 4.10.3 Electric Potential of an Annulus ................................................................. 4-32! 4.10.4 Charge Moving Near a Charged Wire ........................................................ 4-33! 4.10.5! Electric Potential of a Uniformly Charged Sphere ................................... 4-34! 4.11!Conceptual Questions ........................................................................................ 4-35! 4.12!Additional Problems .......................................................................................... 4-36! 4.12.1 Cube ............................................................................................................ 4-36! 4.12.2 Three Charges ............................................................................................. 4-36! 4.12.3 Work Done on Charges............................................................................... 4-36! 4.12.4 Calculating E from V .................................................................................. 4-37! 4-1

4.12.5 Electric Potential of a Rod .......................................................................... 4-37! 4.12.6 Electric Potential ......................................................................................... 4-38! 4.12.7 Calculating Electric Field from the Electric Potential ................................ 4-38! 4.12.8 Electric Potential and Electric Potential Energy ......................................... 4-39! 4.12.9. Electric Field, Potential and Energy .......................................................... 4-39! 4.12.10!P-N Junction.............................................................................................. 4-40! 4.12.11!Sphere with Non-Uniform Charge Distribution ....................................... 4-40! 4.12.12!Electric Potential Energy of a Solid Sphere .............................................. 4-41! 4.12.13!Calculating Electric Field from Electrical Potential ................................. 4-41!

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Electric Potential 4.1 Potential and Potential Energy In the introductory mechanics course, we have seen that force on a particle of mass m located at a distance r from Earth’s center due to the gravitational interaction between the particle and the Earth obeys an inverse-square law:

! Mm Fg = !G 2 rˆ , r

(4.1.1)

where G = 6.67 ! 10"1 1 N # m 2 /kg 2 is the gravitational constant and rˆ is a unit vector pointing radially outward from the Earth. The Earth is assumed to be a uniform sphere of ! mass M. The corresponding gravitation field g , defined as the gravitation force per unit mass, is given by ! GM ! Fg g= = ! 2 rˆ . (4.1.2) m r

! Notice that g is a function of M, the mass that creates the field, and r, the distance from the center of the Earth.

Figure 4.1.1 Consider moving a particle of mass m under the influence of gravity (Figure 4.1.1). The work done by gravity in moving m from A to B is ! ! WG = ) FG ! d s =

# GMm & * G Mm )rA %$ " r 2 (' dr = , rB

+

r

/.

rB

#1 1& = GMm % " ( . $ rB rA '

(4.1.3)

rA

4-3

The result shows that WG is independent of the path taken; it depends only on the endpoints A and B.

! Near Earth’s surface, the gravitational field g is approximately constant, with a magnitude g = GM / rE 2 ! 9.8m/s2 , where rE is the radius of Earth. The work done by gravity in moving an object from height y A to y B (Figure 4.1.2) is yB ! ! Wg = " Fg ! ds = # " mg dy = #mg( y B # y A ) . y

(4.1.4)

A

Figure 4.1.2 Moving an object from A to B. The result again is independent of the path, and is only a function of the change in vertical height y B ! y A . In the examples above, if the object returns to its starting point, then the work done by the gravitation force on the object is zero along this closed path. Any force that satisfies this property for all closed paths is called a conservative force:

""

! ! F!d s = 0

(conservative force).

(4.1.5)

all closed paths

When dealing with a conservative force, it is often convenient to introduce the concept of change in potential energy function, !U = U B " U A between any two points in space, A and B, B! ! (4.1.6) !U = U B " U A = " $ F # d s = "W A

where W is the work done by the force on the object. In the case of gravity, W = Wg and from Eq. ((4.1.3)), the change in potential energy can be written as

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"1 1% U G (rB ) ! U G (rA ) = !GMm $ ! ' # rB rA &

(4.1.7)

It is often convenient to choose a reference point P where U G (rP ) is equal to zero. In the gravitational case, we choose infinity to be the reference point, withU G (rP = !) = 0 . Therefore the change in potential energy when two objects start off an infinite distance apart and end up a distance r apart is given by #1 1 & GMm . U G (r) ! U G (") = !GMm % ! ( = ! r $ r "'

(4.1.8)

Thus we can define a potential energy function U G (r) = !

GMm , r

U (") = 0.

(4.1.9)

When one object is much more massive for example the Earth and a satellite, then the scalar quantity U G (r) , with units of energy, corresponds to the negative of the work done by the gravitation force on the satellite as it moves from an infinite distance away to a distance r from the center of the Earth. The value of U G (r) depends on the choice that U G (rP = !) = 0 . However, the potential energy difference U G (rB ) ! U G (rA ) between two points is independent of the choice of reference point and by definition corresponds to a physical quantity, the negative of the work done.

! Near Earth’s surface, where the gravitation field g is approximately constant, as an object moves from the ground to a height h above the ground, the change in potential energy is !U g = +mgh , and the work done by gravity is Wg = ! mgh . ! Let’s again consider a gravitation field g . Let’s define the change in potential energy per mass between points A and B by !VG " VG (rB ) # VG (rA ) =

U G (rB ) # U G (rA ) !U G " m m

(4.1.10)

According to our definition, B ! B! ! ! !VG = " $ ( FG / m) # d s = " $ g # d s A

A

(4.1.11)

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!VG is called the gravitation potential difference. The terminology is unfortunate because it is very easy to mix-up ‘potential difference’ with ‘potential energy difference’. From Eq. (4.1.7), the gravitation potential difference between the points A and B is B ! B! ! ! !VG = " $ ( FG / m) # d s = " $ g # d s A

A

(4.1.12)

Just like the gravitation field, the gravitation potential difference depends only on the M, the mass that creates the field, and r, the distance from the center of the Earth. Physically !VG represents the negative of the work done per unit mass by gravity to move a particle from points A to B. Our treatment of electrostatics will be similar to gravitation because the electrostatic !" force F e also obeys an inverse-square law. In addition, it is also conservative. In the !" ! presence of an electric field E , in analogy to the gravitational field g , we define the electric potential difference between two points A and B as B ! B ! ! ! !Ve = " $ ( Fe / qt ) # d s = " $ E # d s , A

A

(4.1.13)

where qt is a test charge. The potential difference !Ve , which we will now denote just by !V , represents the negative of the work done per unit charge by the electrostatic force when a test charge qt moves from points A to B. Again, electric potential difference should not be confused with electric potential energy. The two quantities are related as follows. Suppose an object with charge q is moved across a potential difference !V , then the change in the potential energy of the object is

!U = q!V .

(4.1.14)

The SI unit of electric potential is volt [V]

1volt = 1 joule/coulomb (1 V= 1 J/C) .

(4.1.15)

When dealing with systems at the atomic or molecular scale, a joule [J] often turns out to be too large as an energy unit. A more useful scale is electron volt [eV] , which is defined as the energy an electron acquires (or loses) when moving through a potential difference of one volt: (4.1.16) 1eV = (1.6 ! 10"19 C)(1V) = 1.6 ! 10"19 J .

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4.2 Electric Potential in a Uniform Field !" Consider a charge +q moving in the direction of a uniform electric field E = E(! ˆj) , as shown in Figure 4.2.1(a).

(a)

(b)

!" Figure 4.2.1 (a) A charge q moving in the direction of a constant electric field E . (b) A ! mass m moving in the direction of a constant gravitation field g . !" Because the path taken is parallel to E , the electric potential difference between points A and B is given by B !" B " !V = VB " V A = " $ E # d s = " E $ ds = " Ed < 0 . (4.2.1) A

A

Therefore point B is at a lower potential compared to point A. In fact, electric field lines always point from higher potential to lower. The change in potential energy is !U = U B " U A = "qEd . Because q > 0, for this motion !U < 0 , the potential energy of a positive charge decreases as it moves along the direction of the electric field. The corresponding gravity analogy, depicted in Figure 4.2.1(b), is that a mass m loses ! potential energy ( !U = "mgd ) as it moves in the direction of the gravitation field g .

Figure 4.2.2 Potential difference in a uniform electric field

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!" What happens if the path from A to B is not parallel to E , but instead at an angle θ, as shown in Figure 4.2.2? In that case, the potential difference becomes

!" " B !" " !V = VB " V A = " $ E # d s = "E # s = " Es cos % = " Ey . A

(4.2.2)

Note that y increases downward in Figure 4.2.2. Here we see once more that moving !" along the direction of the electric field E leads to a lower electric potential. What would the change in potential be if the path were A ! C ! B ? In this case, the potential difference consists of two contributions, one for each segment of the path: !V = !VCA + !VBC .

(4.2.3)

When moving from A to C, the change in potential is !VCA = " Ey . When moving from C !" to B, !VBC = 0 because the path is perpendicular to the direction of E . Thus, the same !" result is obtained irrespective of the path taken, consistent with the fact that E is a conservative vector field. For the path A ! C ! B , work is done by the field only along the segment AC that is parallel to the field lines. Points B and C are at the same electric potential, i.e.,VB = VC . Because !U = q!V , this means that no work is required when moving the charge from B to C. In fact, all points along the straight line connecting B and C are on the same “equipotential line.” A more complete discussion of equipotential will be given in Section 4.5. 4.3 Electric Potential due to Point Charges Next, let’s compute the potential difference!"between two points A and B due to a charge +Q. The electric field produced by Q is E = (Q / 4!"0 r 2 )ˆr , where rˆ is a unit vector pointing radially away from the location of the charge.

Figure 4.3.1 Potential difference between two points due to a point charge Q. 4-8

! From Figure 4.3.1, we see that rˆ ! d s = dscos" = dr , which gives !V = VB " V A = " %

B A

B Q Q Q ' 1 1* ! ˆ s = " . dr = " r & d %A 4#$ r 2 4 #$0 )( rB rA ,+ 4#$ 0 r 2 0

(4.3.1)

Once again, the potential difference !V depends only on the endpoints, independent of the choice of path taken. As in the case of gravity, only the difference in electrical potential is physically meaningful, and one may choose a reference point and set the potential there to be zero. In practice, it is often convenient to choose the reference point to be at infinity, so that the electric potential at a point P becomes P !" " VP = ! $ E " d s , #

V (#) = 0 .

(4.3.2)

With this choice of zero potential, we introduce an electric potential function, V (r) , where r is the distance from the point-like charged object with charge Q: V (r) =

1 Q . 4!" 0 r

(4.3.3)

When more than one point charge is present, by applying the superposition principle, the electric potential is the sum of potentials due to individual charges: V (r) =

qi q 1 = ke # i # 4!"0 i ri i ri

(4.3.4)

A summary of comparison between gravitation and electrostatics is tabulated below:

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Gravity

Electrostatics

Mass m

Charge q

! Mm Gravitation force FG = !G 2 ˆr r ! ! Gravitation field g = Fg / m

! Qq Electric force Fe = ke 2 rˆ r ! ! Electric field E = Fe / q

B! ! Potential energy change !U = " $ FG # d s

Potential energy change B ! ! !U = " $ Fe # d s

A

A

! ! Gravitational potential !VG = "$ g # d s B

A

Potential function, VG (!) = 0 : VG = !

GM r

! | !U g |= mgd , (constant g )

B! ! Electric Potential !V = " $ E # d s A

Q Potential function, V (!) = 0 : V = ke r !" | !U |= qEd , (constant E)

4.3.1 Potential Energy in a System of Charges Suppose you lift a mass m through a height h. The work done by the external agent (you), is positive, Wext = mgh > 0 . The work done by the gravitation field is negative,

Wg = !mgh = !Wext . The change in the potential energy is therefore equal to the work that you do in lifting the mass, !U g = "Wg = +We xt = mg h . If an electrostatic system of charges is assembled by an external agent, then !U = "W = +We xt . That is, the change in potential energy of the system is the work that must be put in by an external agent to assemble the configuration. The charges are brought in from infinity and are at rest at the end of the process. Let’s start with just two charges q1 and q2 that are infinitely far apart with potential energyU = 0 . Let the potential due to q1 at a point P be V1 (Figure 4.3.2).

Figure 4.3.2 Two point charges separated by a distance r12 . 4-10

The work W2 done by an external agent in bringing the second charge q2 from infinity to P is then W2 = q2V1 . Because V1 = q1 / 4!"0 r12 , where r12 is the distance measured from q1 to P, we have that

U 12 = W2 = q2V1 =

1 q1q2 . 4!"0 r12

(4.3.5)

If q1 and q2 have the same sign, positive work must be done to overcome the electrostatic repulsion and the change in the potential energy of the system is positive, U 12 > 0 . On the other hand, if the signs are opposite, then U 12 < 0 due to the attractive force between the charges. To add a third charge q3 to the system (Figure 4.3.3), the work required is q #q q & W3 = q3 V1 + V2 = 3 % 1 + 2 ( . (4.3.6) 4!"0 $ r13 r23 '

(

)

Figure 4.3.3 A system of three point charges. The potential energy of this configuration is then

U = W2 + W3 =

1 # q1q2 q1q3 q2 q3 & = U 12 + U 13 + U 23 . + + r13 r23 (' 4!" 0 %$ r12

(4.3.7)

The equation shows that the total potential energy is simply the sum of the contributions from distinct pairs. Generalizing to a system of N charges, we have

1 N N qi q j U= # , 4!"0 # i=1 j=1 rij

(4.3.8)

j>i

where the constraint j > i is placed to avoid double counting each pair. Alternatively, one may count each pair twice and divide the result by 2. This leads to

4-11

% 1 N N qi q j 1 N ' 1 N q j U= $ $ = 2 $qi ' 4!" $ r 8!"0 i=1 j =1 rij i=1 0 j =1 ij & j#i j#i

( N * = 1 q V (r ) . $ * 2 i=1 i i )

(4.3.9)

! where V (ri ) , the quantity in the parenthesis is the potential at ri (location of qi ) due to all the other charges.

4.4 Deriving Electric Field from the Electric Potential

!" In Eq. (4.3.2) we established the relation between E and V. If we consider two points that ! are separated by a small distance d s , the following differential form is obtained: !" " dV = !E " d s .

(4.4.1)

!" ˆ and d s! = dx ˆi + dyˆj + dz k, ˆ and therefore In Cartesian coordinates, E = Ex ˆi + E y ˆj + Ez k

ˆi + dyˆj + dz k) ˆ ˆ = E dx + E dy + E dz . dV = (Ex ˆi + E yjˆ + Ez k!)(dx x y z

(4.4.2)

We define directional derivatives !V / !x , !V / !y , and ! V / ! z such that

!V !V !V dy + dx + dz . !y !x !z

(4.4.3)

"V "V "V , Ey = ! , Ez = ! . "x "y "z

(4.4.4)

dV = Therefore

Ex = !

By introducing a differential quantity called the del (gradient) operator

!"

#ˆ # ˆ # ˆ j+ k i+ #z #x #y

(4.4.5)

the electric field can be written as !" # "V ˆ "V ˆ "V ˆ & k = !)V . E = Ex ˆi + E y ˆj + Ez kˆ = ! % j+ i+ "z (' "y $ "x

(4.4.6)

The differential operator, ! , operates on a scalar quantity (electric potential) and results !" in a vector quantity (electric field). Mathematically, we can think of E as the negative of the gradient of the electric potential V . Physically, the negative sign implies that if V increases as a positive charge moves along some direction, say x, with !V / !x > 0 , then 4-12

!" there is a non-vanishing component of E in the opposite direction Ex = !"V / "x < 0 . In the case of gravity, if the gr...