Gauss's Law for Electric Field

The net electric ﬂux ΦE through any closed surface is equal to the net charge Qin inside divided by the permittivity constant ǫ0:

Qin Qin −12 2 −1 −2 E~ · dA~ = 4πkQin = i.e. ΦE = with ǫ0 = 8.854 × 10 C N m ǫ ǫ I 0 0 The closed surface can be real or ﬁctitious. It is called “Gaussian surface”. The symbol denotes an integral over a closed surface in this context. H

• Gauss’s law is a general relation between electric charge and electric ﬁeld. • In electrostatics: Gauss’s law is equivalent to Coulomb’s law. • Gauss’s law is one of four Maxwell’s equations that govern cause and effect in electricity and magnetism.

1/9/2015 [tsl44 – 1/32] Gauss’ Law for Electric Field (Illustration)

1/9/2015 [tsl445 – 2/32] Gaussian surface problem (1)

(A) The electric ﬂuxes through the Gaussian surfaces SA and SB are ΦE = 5C/ǫ0 and (B) ΦE = 3C/ǫ0, respectively.

SB SA

q1 q2

Find the electric charges q1 and q2.

1/9/2015 [tsl45 – 3/32] Gaussian surface problem (2)

(A) The electric ﬂuxes through the Gaussian surfaces SA and SB are ΦE = 1C/ǫ0 and (B) ΦE = 3C/ǫ0, respectively.

SA SB

q q1 =2C 2 q3

Find the electric charges q2 and q3.

1/9/2015 [tsl46 – 4/32] Gaussian surface problem (3)

A proton, a neutron, and an electron are placed in different boxes. The electric ﬂuxes through the three Gaussian surfaces are as indicated.

ε ε e/ 0 e/ 0 zero A BC

Name the particle in each box.

1/9/2015 [tsl47 – 5/32] Gaussian surface problem (4)

Three point charges q1, q2, q3 produce electric ﬂuxes through the three Gaussian surfaces as indicated.

ε ε 2C/ 0 1C/ 0 q 2

q q 1 3

ε 5C/ 0

(a) Find the net charge Q = q1 + q2 + q3.

(b) Find the individual charges q1, q2, q3.

1/9/2015 [tsl48 – 6/32] Gaussian surface problem (5)

A positive charge and a negative charge are placed in different boxes. One box remains empty. The electric ﬂuxes through the three Gaussian surfaces are as indicated.

ε ε −4C/ 0 +2C/ 0 A

B C

ε −2C/ 0

(a) Which box contains the positive charge? (b) Which box contains the negative charge?

1/9/2015 [tsl49 – 7/32] Gaussian surface problem (6)

(A) The electric ﬂuxes through the Gaussian surfaces SA, SB, and SC are ΦE = 4C/ǫ0, (B) (C) ΦE = 1C/ǫ0, and ΦE = 2C/ǫ0, respectively.

SA SB

q q1 2 q3

Find the electric charges q1, q2, and q3.

1/9/2015 [tsl50 – 8/32] Calculating E~ from Gauss’s Law: Strategy

Design the Gaussian surface such that it reﬂects the symmetry of the problem at hand.

• Use concentric Gaussian spheres in problems with spherically symmetric charge distributions. The electric field is perpendicular to the Gaussian sphere (E~ k dA~). • Use coaxial Gaussian cylinders in problems with cylindrically symmetric charge distributions. The electric field is perpendicular to the curved surface (E~ k dA~) and parallel to the flat surfaces (E~ ⊥ dA~). • Use Gaussian cylinders with axis perpendicular to planar charge distributions. The electric field is parallel to the curved surface (E~ ⊥ dA~) and perpendicular to the flat surfaces (E~ k dA~).

1/9/2015 [tsl51 – 9/32] Calculating E~ from Gauss’s Law: Point Charge

• Consider a positive point charge Q. • Use a Gaussian sphere of radius R centered at the location of Q. • Surface area of sphere: A = 4πR2.

2 • Electric ﬂux through Gaussian surface: ΦE = E~ · dA~ = E(4πR ). I • Net charge inside Gaussian surface: Qin = Q.

Qin Q • Gauss’s law E~ · dA~ = becomes E(4πR2)= . ǫ ǫ I 0 0 1 Q kQ • Electric ﬁeld at radius R: E = = . 2 2 4πǫ0 R R

1/9/2015 [tsl52 – 10/32] Calculating E~ from Gauss’s Law: Charged Wire

• Consider a uniformly charged wire of inﬁnite length. • Charge per unit length on wire: λ (here assumed positive). • Use a coaxial Gaussian cylinder of radius R and length L.

• Electric ﬂux through Gaussian surface: ΦE = E~ · dA~ = E(2πRL). I • Net charge charge inside Gaussian surface: Qin = λL.

Qin λL • Gauss’s law E~ · dA~ = becomes E(2πRL)= . ǫ ǫ I 0 0 1 λ • Electric ﬁeld at radius R: E = . 2πǫ0 R

1/9/2015 [tsl53 – 11/32] Calculating E~ from Gauss’s Law: Charged Plane Sheet

• Consider a uniformly charged plane sheet. • Charge per unit area on sheet: σ (here assumed positive). • Use Gaussian cylinder with cross-sectional area A placed as shown.

• Electric ﬂux through Gaussian surface: ΦE = E~ · dA~ = 2EA. I Net charge charge inside Gaussian surface: Qin = σA.

Qin σA • Gauss’s law E~ · dA~ = becomes 2EA = . ǫ ǫ I 0 0 σ • Electric ﬁeld at both ends of cylinder: E = 2ǫ0 (pointing away from sheet). • Note that E does not depend on the distance from the sheet.

1/9/2015 [tsl54 – 12/32] Calculating E~ from Gauss’s Law: Charged Slab

• Consider a uniformly charged slab. • Charge per unit volume on slab: ρ. • Use Gaussian cylinder as shown.

• Total electric ﬂux: ΦE = 2|Ez|A.

2ρA|z| (|z|≤ a) • Net charge inside: Qin = ( 2ρAa (|z|≥ a)

2ρA|z| ǫ0 (|z|≤ a) • Gauss’s law: 2|Ez|A = 2ρAa ( ǫ0 (|z|≥ a)

ρa − ǫ0 (z ≤−a) • Electric ﬁeld: E = ρz (−a ≤ z ≤ a) z 8 ǫ0 > ρa < ǫ0 (z ≥ a) :>

1/9/2015 [tsl373 – 13/32] Electric Field of Uniformly Charged Spherical Shell

• Radius of charged spherical shell: R • Electric charge on spherical shell: Q = σA = 4πσR2. • Use a concentric Gaussian sphere of radius r. Q • r > R: E(4πr2)= ǫ0 1 Q ⇒ E = 2 4πǫ0 r Qin • r < R: E(4πr2)= = 0 ǫ0 ⇒ E = 0

1/9/2015 [tsl55 – 14/32] Electric Field of Uniformly Charged Solid Sphere

• Radius of charged solid sphere: R • Electric charge on sphere: 4π Q = ρV = ρR3. 3 • Use a concentric Gaussian sphere of radius r. Q • r > R: E(4πr2)= ǫ0 1 Q ⇒ E = 2 4πǫ0 r 1 4π • r < R: E(4πr2)= r3ρ ǫ 3 0 „ « ρ 1 Q ⇒ E(r)= r = r 3 3ǫ0 4πǫ0 R

1/9/2015 [tsl56 – 15/32] Electric Field of Oppositely Charged Inﬁnite Sheets

• Consider two infinite sheets of charge with charge per unit area ±σ, respectively. • The sheets are positioned at x = 0 and x = 2m, respectively. σ • Magnitude of field produced by each sheet: E = . 2ǫ0 (+) (−) σ σ • Electric field at x< 0: Ex = Ex + Ex = − + = 0. 2ǫ0 2ǫ0 (+) (−) σ σ σ • Electric field at 0 2m: Ex = Ex + Ex =+ − = 0. 2ǫ0 2ǫ0

1/9/2015 [tsl57 – 16/32] Electric Field Near Charged Inﬁnite Sheets

Consider three pairs of parallel, inﬁnite, uniformly charged sheets. The charge per unit area is equal in magnitude on all sheets. Find the direction (↑, ↓, none) of the electric ﬁeld at the nine locations indicated.

E1 E 4 E 7

σ > 0 σ < 0 σ > 0

E 2 E 5 E 8

σ > 0 σ < 0 σ < 0

E E 3 E 6 9

1/9/2015 [tsl446 – 17/32] Unit Exam I: Problem #2 (Spring ’09)

Consider two very large uniformly charged parallel sheets as shown. The charge densities are −12 −2 −12 −2 σA = +7 × 10 Cm and σB = −4 × 10 Cm , respectively. Find magnitude and direction (left/right) of the electric ﬁelds E1, E2, and E3.

EE1 2 E3 σ Α σΒ

1/9/2015 [tsl390 – 18/32] Unit Exam I: Problem #2 (Spring ’09)

EE1 2 E3 Solution: σ Α σΒ

|σA| EA = = 0.40N/C (directed away from sheet A). 2ǫ0

1/9/2015 [tsl390 – 18/32] Unit Exam I: Problem #2 (Spring ’09)

EE1 2 E3 Solution: σ Α σΒ

|σA| EA = = 0.40N/C (directed away from sheet A). 2ǫ0

|σB| EB = = 0.23N/C (directed toward sheet B). 2ǫ0

1/9/2015 [tsl390 – 18/32] Unit Exam I: Problem #2 (Spring ’09)

EE1 2 E3 Solution: σ Α σΒ

|σA| EA = = 0.40N/C (directed away from sheet A). 2ǫ0

|σB| EB = = 0.23N/C (directed toward sheet B). 2ǫ0

E1 = EA − EB = 0.17N/C (directed left).

1/9/2015 [tsl390 – 18/32] Unit Exam I: Problem #2 (Spring ’09)

EE1 2 E3 Solution: σ Α σΒ

|σA| EA = = 0.40N/C (directed away from sheet A). 2ǫ0

|σB| EB = = 0.23N/C (directed toward sheet B). 2ǫ0

E1 = EA − EB = 0.17N/C (directed left).

E2 = EA + EB = 0.63N/C (directed right).

1/9/2015 [tsl390 – 18/32] Unit Exam I: Problem #2 (Spring ’09)

EE1 2 E3 Solution: σ Α σΒ

|σA| EA = = 0.40N/C (directed away from sheet A). 2ǫ0

|σB| EB = = 0.23N/C (directed toward sheet B). 2ǫ0

E1 = EA − EB = 0.17N/C (directed left).

E2 = EA + EB = 0.63N/C (directed right).

E2 = EA − EB = 0.17N/C (directed right).

1/9/2015 [tsl390 – 18/32] Unit Exam I: Problem #2 (Spring ’12)

Two very large, thin, uniformly charged, parallel sheets are positioned as shown. Find the values of the charge densities (charge per area), σA and σB, if you know the electric ﬁelds E1, E2, and E3.

Consider two situations. (a) E1 = 2N/C (directed left), E2 = 0, E3 = 2N/C (directed right). (b) E1 = 0, E2 = 2N/C (directed right), E3 = 0.

EE1 2 E3 σ Α σΒ

1/9/2015 [tsl423 – 19/32] Unit Exam I: Problem #2 (Spring ’12)

Consider two situations. (a) E1 = 2N/C (directed left), E2 = 0, E3 = 2N/C (directed right). (b) E1 = 0, E2 = 2N/C (directed right), E3 = 0.

EE1 2 E3 σ Solution: Α σΒ (a) The two sheets are equally charged: −11 2 σA = σB = 2ǫ0(1N/C)=1.77 × 10 C/m .

1/9/2015 [tsl423 – 19/32] Unit Exam I: Problem #2 (Spring ’12)

Consider two situations. (a) E1 = 2N/C (directed left), E2 = 0, E3 = 2N/C (directed right). (b) E1 = 0, E2 = 2N/C (directed right), E3 = 0.

EE1 2 E3 σ Solution: Α σΒ (a) The two sheets are equally charged: −11 2 σA = σB = 2ǫ0(1N/C)=1.77 × 10 C/m . (b) The two sheets are oppositely charged: −11 2 σA = −σB = 2ǫ0(1N/C) = 1.77 × 10 C/m .

1/9/2015 [tsl423 – 19/32] Charged Conductor at Equilibrium (1)

• Consider a conductor with excess charge Q in isolation. • The mobile charges (electrons) are rearranged

spontaneously until we have E~0 = 0 everywhere inside the conductor. • If E~ = 0 inside the conductor, then Gauss’s 0 Q law implies that there can be no net ﬂux through any Gaussian surface that is inside the E0 (r) conductor. E0 = 0 • Hence there can be no net charge in any region inside the conductor. • Hence all excess charge must be at the surface, where it produces an electric ﬁeld

E~0(~r) on the outside only.

1/9/2015 [tsl58 – 20/32] Charged Conductor at Equilibrium (2)

• Now place a point charge q near the charged conductor. • The electric ﬁeld produced by q causes a further rearrangement of mobile surface charges until we have again E~ = 0 in the interior. • Locally, the electric ﬁeld E~ is perpendicular to the surface of the conductor, and its magnitude is proportional to the charge per unit area: E = σ/ǫ0.

E(r) E = 0 + q + + + ++ + +

1/9/2015 [tsl59 – 21/32] Charged Conductor at Equilibrium (3)

• Consider a conductor with a cavity and excess charge Q. • Gauss’s law implies that there is no net charge on the surface of the cavity.

• The external field is E~0(~r). There is no field in the cavity. • Now place a point charge q inside the cavity. • Gauss’s law implies that there is a charge −q on the surface of the cavity. • Charge conservation implies that there is a charge Q + q on the outer surface of the conductor. • The external field changes to E~ (~r). There is a nonzero electric field field inside the cavity.

Q Q+q

−q + E = 0 q

E = 0 E = 0

E0(r) E(r)

1/9/2015 [tsl60 – 22/32] Charged Conductor Problem (1)

Consider a metal cube with a charge 2C on it positioned inside a cubic metal shell with a charge −1C on it.

• Find the charge Qint on the interior surface and the charge Qext on the exterior surface of the shell.

−1C

2C Q int Q ext

1/9/2015 [tsl61 – 23/32] Charged Conductor Problem (2)

A conducting spherical shell of inner radius r1 = 4cm and outer radius r2 = 6cm carries no net charge. Now we place a point charge q = −1µC at its center.

(a) Find the surface charge densities σ1 and σ2.

(b) Find the electric ﬁelds E1 and E2 in the immediate vicinity of the shell. (c) What happens to the electric ﬁelds inside and outside the shell when a second point charge Q = +1µC is placed a distance d = 20cm from the center of the shell? (d) Which objects exert a force on the second point charge?

r1 r 2 Q q + q1 q 2

1/9/2015 [tsl62 – 24/32] Charged Conductor Problem (3)

A point charge qp = −7µC is positioned at the center of a conducting spherical shell with a charge qs = +4µC on it.

• Find the direction (inward/outward) of the electric ﬁeld at the points A and B. int ext • Find the charge qs on the inner surface and the charge qs on the outer surface of the shell.

qp A B

1/9/2015 [tsl63 – 25/32] Charged Conductor Problem (4)

A long conducting cylinder of radius R0 = 3cm carries a charge per unit length λc = 5.0µC/m. It is surrounded by a conducting cylindrical shell of radii R1 = 7cm and R2 = 11cm. The shell carries a charge per unit length λs = −8.0µC/m.

(a) Find the linear charge densities λ1, λ2 on the inner and outer surfaces of the shell.

(b) Find the electric ﬁelds E0, E1, E3 in the vicinity of the three conducting surfaces.

R1 R2 R0

1/9/2015 [tsl64 – 26/32] Charged Conductor Problem (5)

Consider two concentric shells with charges on them as indicated.

• Find the charges qA, qB, qC , qD on each of the four surfaces.

−3C D C B A

1/9/2015 [tsl65 – 27/32] Intermediate Exam I: Problem #4 (Spring ’05)

Consider two concentric conducting spherical shells. The total electric charge on the inner shell is 4C and the total electric charge on the outer shell is −3C. Find the electric charges q1, q2, q3, q4 on each surface of both shells as identiﬁed in the ﬁgure.

−3C

4C 4 3 2 1