Flux (1)

FluxFlux

How do we describe the motion of How do we describe the motion of air through a window?air through a window?

Flux ofFlux of EE

How does E pass How does E pass through a random through a random surface?surface?

22-2722-27

A square that has 10-cm-long edges is A square that has 10-cm-long edges is centered on the centered on the xx axis in a region where there axis in a region where there exists a uniform electric field given by exists a uniform electric field given by

((aa) What is the electric flux of this electric field ) What is the electric flux of this electric field through the surface of a square if the normal through the surface of a square if the normal to the surface is in the + to the surface is in the + xx direction? ( direction? (bb) What ) What is the electric flux through the same square is the electric flux through the same square surface if the normal to the surface makes a surface if the normal to the surface makes a 60° angle with the 60° angle with the yy axis and an angle of 90° axis and an angle of 90° with the with the zz axis? axis?

A cubical surface with no charge enclosed and A cubical surface with no charge enclosed and with sides 2.0 m long is oriented with right and with sides 2.0 m long is oriented with right and

left faces perpendicular to a uniform electric field left faces perpendicular to a uniform electric field EE of (1.6 of (1.6 10 1055 N/C) N/C) in the +x direction. The net in the +x direction. The net

electric flux electric flux EE through this surface is through this surface is

approximatelyapproximately

A.A. zerozero

B.B. 6.4 6.4 10 1055 N· N· mm22/C/C

C.C. 13 13 10 1055 N· m N· m22/C/C

D.D. 25 25 10 1055 N· m N· m22/C/C

E.E. 38 38 10 1055 N· m N· m22/C/C

ApplicationsApplications

Surface 1Surface 1 Surface 2Surface 2 Surface 3Surface 3

Flux = 0???Flux = 0??? Surface 4Surface 4

Flux = 0???Flux = 0???

A surface is so constructed that, at all A surface is so constructed that, at all points on the surface, the points on the surface, the EE vector vector points inward. Therefore, it can be points inward. Therefore, it can be

said that said that

A.A. the surface encloses a net the surface encloses a net positive charge. positive charge.

B.B. the surface encloses a net the surface encloses a net negative charge. negative charge.

C.C. the surface encloses no net the surface encloses no net charge. charge.

Gauss’s LawGauss’s Law

Gauss’s law relates the net flux Gauss’s law relates the net flux ΦΦ of of an electric field through a closed an electric field through a closed surface (a Gaussian surface) to the surface (a Gaussian surface) to the netnet charge charge qqencenc that is that is enclosedenclosed by by that surface. that surface.

Read section 22-2

Find the electric field due to a Find the electric field due to a uniformly charged thin spherical uniformly charged thin spherical shell of radius R and total charge Qshell of radius R and total charge Q

ReminderReminder

Gauss’s law relates the net flux Gauss’s law relates the net flux ΦΦ of of an electric field through a closed an electric field through a closed surface (a Gaussian surface) to the surface (a Gaussian surface) to the netnet charge charge qqencenc that is that is enclosedenclosed by by that surface. that surface.

Read section 22-2

22-3022-30

Careful measurement of the electric Careful measurement of the electric field at the surface of a black box field at the surface of a black box indicates that the net outward electric indicates that the net outward electric flux through the surface of the box is flux through the surface of the box is

((aa) What is the net charge inside the ) What is the net charge inside the box? (box? (bb) If the net outward electric ) If the net outward electric flux through the surface of the box flux through the surface of the box were zero, could you conclude that were zero, could you conclude that there were no charges inside the box? there were no charges inside the box? Explain your answer.Explain your answer.

The figure shows a surface enclosing The figure shows a surface enclosing the charges 2the charges 2qq and – and –qq. The net flux . The net flux

through the surface surrounding the two through the surface surrounding the two charges is charges is

3

E.

zero D.

C.

2 B.

A.

0

0

0

0

q

q

ε

q

q

The figure shows a surface The figure shows a surface enclosing the charges enclosing the charges qq and – and –qq. . The net flux through the surface The net flux through the surface surrounding the two charges is surrounding the two charges is

A.A. qq//00

B.B. 22qq//00

C.C. – –qq//00

D.D. zero zero

E.E. – –22qq//00

22-3322-33

A single point charge is placed at the A single point charge is placed at the center of an imaginary cube that has center of an imaginary cube that has 20-cm-long edges. The electric flux 20-cm-long edges. The electric flux out of one of the cube’s sides isout of one of the cube’s sides is

How much charge is at the center? How much charge is at the center?

Ans -79.9 nc

More AppsMore Apps

CylindersCylinders PlanesPlanes SpheresSpheres

Look for the symmetry and exploit it! Look for the symmetry and exploit it!

22-3822-38

A nonconducting thin spherical shell A nonconducting thin spherical shell of radius 6.00 cm has a uniform of radius 6.00 cm has a uniform surface charge density of 9.00 surface charge density of 9.00 nC/mnC/m22. (. (aa) What is the total charge ) What is the total charge on the shell? Find the electric field on the shell? Find the electric field at the following distances from the at the following distances from the sphere’s center: (sphere’s center: (bb) 2.00 cm, () 2.00 cm, (cc) 5.90 ) 5.90 cm, (cm, (dd) 6.10 cm, and () 6.10 cm, and (ee) 10.0 cm. ) 10.0 cm.

22-4422-44

A sphere of radius A sphere of radius RR has volume has volume charge density charge density ρρ = = CC//rr22 for for rr < < RR, , where where CC is a constant and is a constant and ρρ = 0 for = 0 for rr > > RR. (. (aa) Find the total charge on the ) Find the total charge on the sphere. (sphere. (bb) Find the expressions for the ) Find the expressions for the electric field inside and outside the electric field inside and outside the charge distribution. (charge distribution. (cc) Sketch the ) Sketch the magnitude of the electric field as a magnitude of the electric field as a function of the distance function of the distance rr from the from the sphere’s center.sphere’s center.

22-44 pix22-44 pix

22-4822-48

Show that the electric field due to an Show that the electric field due to an infinitely long, uniformly charged infinitely long, uniformly charged thin cylindrical shell of radius thin cylindrical shell of radius a a having a surface charge density σ is having a surface charge density σ is given by the following expressions: given by the following expressions: EE = 0 for = 0 for

22-5022-50

An infinitely long nonconducting An infinitely long nonconducting solid cylinder of radius solid cylinder of radius a a has a has a uniform volume charge density of ρuniform volume charge density of ρ00. . Show that the electric field is given Show that the electric field is given by the following expressions: by the following expressions:

for for

where where RR is the distance from the long is the distance from the long axis of the cylinder. axis of the cylinder.

Qra

rb1

rb2

A solid conducting sphere of radius A solid conducting sphere of radius rraa is placed is placed

concentrically inside a conducting spherical shell of concentrically inside a conducting spherical shell of inner radius inner radius rrb1b1 and outer radius and outer radius rrb2b2. The inner sphere . The inner sphere

carries a charge Q while the outer sphere does not carry carries a charge Q while the outer sphere does not carry any net charge. The electric field for any net charge. The electric field for rra a rr rrb1b1 is is

ero E.

ˆ2

D.

ˆ2

C.

ˆ B.

ˆ A.

2

2

z

rr

kQ

rr

kQ

rr

kQ

rr

kQ

E discontinuityE discontinuity

E is discontinuous E is discontinuous for a spherical-shell for a spherical-shell distributiondistribution

E discontinuityE discontinuity

E is NOT discontinuous at the E is NOT discontinuous at the perimeter of a solid sphere of charge perimeter of a solid sphere of charge (finite volume charge density)(finite volume charge density)

Charge and Field at the Charge and Field at the surface of a conductorsurface of a conductor

Conductor: charges move “easily”Conductor: charges move “easily” So at equilibrium, E inside a So at equilibrium, E inside a

conductor is zero everywhereconductor is zero everywhere

Read section 22-5Read section 22-5

A solid conducting sphere of radius A solid conducting sphere of radius rraa is placed is placed

concentrically inside a conducting spherical shell of concentrically inside a conducting spherical shell of inner radius inner radius rrb1b1 and outer radius and outer radius rrb2b2. The inner sphere . The inner sphere

carries a charge Q while the outer sphere does not carry carries a charge Q while the outer sphere does not carry any net charge. The electric field for any net charge. The electric field for rrb1 b1 rr rrb2b2 is is

Qra

rb1

rb2

ero E.

ˆ2

D.

ˆ2

C.

ˆ B.

ˆ A.

2

2

z

rr

kQ

rr

kQ

rr

kQ

rr

kQ

A solid conducting sphere of radius A solid conducting sphere of radius rraa is placed is placed

concentrically inside a conducting spherical shell of concentrically inside a conducting spherical shell of inner radius inner radius rrb1b1 and outer radius and outer radius rrb2b2. The inner sphere . The inner sphere

carries a charge Q while the outer sphere does not carry carries a charge Q while the outer sphere does not carry any net charge. The electric field for any net charge. The electric field for rr rrb2b2 is is

ero E.

ˆ2

D.

ˆ2

C.

ˆ B.

ˆ A.

2

2

z

rr

kQ

rr

kQ

rr

kQ

rr

kQ

Qra

rb1

rb2

22-1722-17

A ring that has radius A ring that has radius aa lies in the lies in the zz = 0 = 0 plane with its center at the origin. The plane with its center at the origin. The ring is uniformly charged and has a ring is uniformly charged and has a total charge total charge QQ. Find . Find EzEz on the on the zz axis at axis at ((aa) ) zz = 0.2 = 0.2aa, (, (bb) ) zz = 0.5 = 0.5aa, (, (cc) ) zz = 0.7 = 0.7aa, , ((dd) ) zz = = aa, and (, and (ee) ) zz = 2 = 2aa. (. (ff) Use your ) Use your results to plot results to plot EzEz versus versus zz for both for both positive and negative values of positive and negative values of zz (Assume that these distances are (Assume that these distances are exact.) exact.)

22-17 pix22-17 pix

22-2322-23

A line of charge that has uniform A line of charge that has uniform linear charge density linear charge density λλ lies on the lies on the xx axis from axis from xx = 0 to = 0 to xx = = aa. Show that . Show that the the yy component of the electric field component of the electric field at a point on the at a point on the yy axis is given by axis is given by

22-3122-31 A point charge A point charge qq = 2.0 = 2.0 μμC is at the center of C is at the center of

an imaginary sphere that has a radius equal to an imaginary sphere that has a radius equal to 0.500 m. (0.500 m. (aa) Find the surface area of the ) Find the surface area of the sphere. (sphere. (bb) Find the magnitude of the electric ) Find the magnitude of the electric field at all points on the surface of the sphere. field at all points on the surface of the sphere. ((cc) What is the flux of the electric field through ) What is the flux of the electric field through the surface of the sphere? (the surface of the sphere? (dd) Would your ) Would your answer to Part (answer to Part (cc) change if the point charge ) change if the point charge were moved so that it was inside the sphere were moved so that it was inside the sphere but not at its center? (but not at its center? (ee) What is the flux of the ) What is the flux of the electric field through the surface of an electric field through the surface of an imaginary cube that has 1.00-m-long edges imaginary cube that has 1.00-m-long edges and encloses the sphere? and encloses the sphere?

22-6822-68

An infinitely long line charge that An infinitely long line charge that has a uniform linear charge density has a uniform linear charge density equal to -1.5equal to -1.5μμC/m lies parallel to the C/m lies parallel to the yy axis at axis at xx = -2.00 m. A positive = -2.00 m. A positive point charge that has a magnitude point charge that has a magnitude equal to 1.30equal to 1.30μμC is located at C is located at xx = = 1.00 m, 1.00 m, yy = 2.00 m. Find the electric = 2.00 m. Find the electric field at field at xx = 2.00 m, = 2.00 m, yy = 1.50 m. = 1.50 m.

1.62kN/C i - 4.18kN/C j

22-7722-77 An infinite nonconducting plane sheet of charge An infinite nonconducting plane sheet of charge

that has a surface charge density +3.00that has a surface charge density +3.00μμC/mC/m22 lies in the lies in the yy = -0.600 m plane. A second infinite = -0.600 m plane. A second infinite nonconducting plane sheet of charge that has a nonconducting plane sheet of charge that has a surface charge density of surface charge density of -2.00-2.00μμC/mC/m22 lies in the lies in the xx = 1.00 m plane. Lastly, a = 1.00 m plane. Lastly, a nonconducting thin spherical shell that has a nonconducting thin spherical shell that has a radius of 1.00 m and its center in the radius of 1.00 m and its center in the zz = 0 plane = 0 plane at the intersection of the two charged planes has at the intersection of the two charged planes has a surface charge density of -3.00a surface charge density of -3.00μμC/mC/m22. .

Find the magnitude and direction of the electric Find the magnitude and direction of the electric field on the field on the xx axis at ( axis at (aa) ) xx = 0.400 m and ( = 0.400 m and (bb) ) xx = = 2.50 m. 2.50 m.