AP Physics C - E & M

AP Physics C - E & M Gauss's Law 2017-07-08 www.njctl.org

Electric Flux Gauss's Law Sphere Table of Contents: Gauss's Law Click on the topic to go to that section. Infinite Rod of Charge Infinite Plane of Charge Electrostatic Equilibrium of Conductors Move any photo or image in this presentation to reveal a link to its source, providing attribution and additional information.

Electric Flux Return to Table of Contents

Electric Flux An Electric Field,, represented by the Electric Field lines below, passes through a rectangular area of space, with cross sectional area, A. The field is perpendicular to the area. Define Electric Flux as the strength of the Electric Field times the area through which it passes: Visually, we use the number of field lines to represent the Electric Field strength. The more lines passing through an area of space - the greater the Electric Flux.

Electric Flux But what if the Electric Field is not perpendicular to the area of space that we're interested in finding out the Electric Flux? Find the Electric Flux through the slanted area of the below shape. A is the area of the slanted side and A 1 is the area of the vertical side. The Electric Field makes an angle θ with the vector da which is normal (perpendicular) to the light blue surface area.

Electric Flux The number of field lines through surfaces A 1 and A is the same. The widths of each surface are the same, and are denoted by. The lengths of each surface are denoted by and. These lengths are two sides of a right triangle and related by: The areas of the two surfaces:

Electric Flux Since the flux through both surfaces is the same: The flux is proportional to the number of field lines, and the cosine of the angle that the normal to the surface makes with the field.

Electric Flux Not all surfaces are ramps. Let's see how calculus will help in calculating the flux through an arbitrary shape. Consider the 3 dimensional shape to the left. It is bounded by a "closed surface," which means it has an inside and an outside on either side of the surface boundary. You need to go through the boundary to switch sides.

Electric Flux The flux through any area element ΔA i of the closed surface is: To find the total flux through this surface, shrink ΔA i to almost zero and sum all the ΔΦ Ei 's over the entire surface: Note the different symbol for the integral - that circle means you are integrating over an entire closed surface - it is called a closed surface integral. Sometimes a double integral sign is used.

1 A rectangular loop is held perpendicular to a uniform Electric Field (the loop's normal vector,da, is parallel to the Electric Field direction). At what angle to its normal does the loop have to be turned through so the electric flux is one half of its original value? A 15 B 30 C 45 D 60 E 90 o o o o o Answer

2 The normal vector to a rectangular loop makes an angle of 35 0 with the vector describing the direction of an applied electric field. What is the Electric Flux through the loop, given the following data? A 1.1 x 10 4 Nm 2 /C B 7.8 x 10 3 Nm 2 /C C 1.1 x 10 8 Nm 2 /C Answer D 7.8 x 10 7 Nm 2 /C E Zero

Electric Flux through a Closed Surface The closed surface definition will allow us to use a very powerful tool to quantify Electric Fields due to charge distributions. For Electric Field lines passing through a closed surface, we define a sign convention: Field lines going from the outside to the inside of a closed surface are given a negative sign. Field lines going from inside to the outside of a closed surface are given a positive sign. The net Electric Flux through a closed surface is then equal to the number of field lines leaving minus the number of lines entering.

Charge and Electric Flux Here are three identical boxes with different combinations of charges within their closed surfaces. Each charge has the same magnitude and generates an electric field represented by the red field lines. - - Box A Box B What is the Electric Flux through each box? Box C

Charge and Electric Flux - - Box A Box B Box C Box A: The electric field lines point out of the box, therefore it has an outward electric flux (positive). Box B: The electric field lines point into the surface of the box, therefore it has an inward electric flux (negative). Box C: Has a net charge of zero within the box, and since the flow is the same but in opposite directions for each charge, the electric flux is zero.

Charge and Electric Flux An empty box is placed in an external Electric field. The amount of charge enclosed by the box is zero. The net electric flux is zero because the number of field lines entering is equal to the number of field lines leaving.

Charge and Electric Flux q 2q If the electric field doubles, then the number of field lines representing the field through the surface will also double. From this we can conclude that the net electric flux is directly proportional to the enclosed charge.

Charge and Electric Flux A positive charge is surrounded by two concentric spheres - what is the relationship of the electric flux through each sphere?

Charge and Electric Flux The Electric Flux through each sphere is the same. As the distance from the charge increases, the Electric Field decreases as 1/r 2, while the surface area that encloses the field lines increases as a factor of r 2. Since Electric Flux is a product of the field strength and the surface area, these factors cancel out. The Electric Flux is independent of the size or shape of the enclosing surface.

3 A positive charge is enclosed by a spherical surface. What direction is the electric flux relative to the surface? A Tangent. B Always perpendicular. C From the outside to the inside of the surface. Answer D From the inside to the outside of the surface. E The lines do not pass through the surface.

4 A charged particle is enclosed within a rectangular box and produces an Electric Flux of Φ through the box. What is the Electric Flux if the dimensions of the box are tripled? A 1/9 Φ B 1/3 Φ C Φ Answer D 3 Φ E 9 Φ

Gauss's Law Return to Table of Contents

Gauss's Law Electric Flux will now be used to derive Gauss's Law. Start with a positive charge, q, and surround it with a hypothetical spherical surface - called a "Gaussian surface." This surface has no physical meaning - it's being used for its symmetry properties, and is typically shown as a dashed line or a shaded surface. Gaussian surface

Gauss's Law For any segment on the surface of the sphere, ΔA i, its normal vector, da i and the Electric Field, E are parallel. Furthermore, E is constant everywhere on the surface. The flux through this segment is: The total flux through the Gaussian surface is then:

Gauss's Law The magnitude of E at a distance r from a point charge is: Substitute this value in the flux equation found on the previous slide: This is one of the reasons that ε 0 is used instead of κ - we get rid of 4π in Gauss's Law.

Gauss's Law One more step - to show that works for any closed surface surrounding a charge. All three surfaces have the same Electric Flux through them - visually, the same amount of Electric Field lines go through each surface. Since they all enclose the same charge, the above equation works, regardless of the shape of the enclosing surface.

Gauss's Law If there is more than one charge enclosed by the surface, you just need to add the charges algebraically and Gauss's Law applies. This is a powerful equation, but several assumptions are required in order to use it to solve problems using simple integrals. Here's the list: The Electric field is constant (which could also equal zero) at certain segments of the chosen Gaussian surface. The Electric field is perpendicular to the surface so that. or, The Electric field is parallel to the surface so that.

Charge Density When finding the Electric field due to a charge distribution, the following definitions (from last chapter) will be useful. Linear charge density: Surface charge density: Volume charge density:

5 A charge of magnitude q, is at the center of a sphere of radius r. What is the electric flux at the surface of the sphere? A 5.6x10 5 Nm 2 /C B 6.5x10 5 Nm 2 /C C 7.3x10 5 Nm 2 /C Answer D 1.1x10 6 Nm 2 /C E 1.3x10 6 Nm 2 /C

6 You are trying to calculate the Electric Field outside a positively charged metal sphere of radius R. The Gaussian surface should be what kind of geometrical object? A Circle B Cylinder C Cube R Answer D Sphere E Rectangle

7 You are trying to calculate the Electric Field outside a positive line of charge. The Gaussian surface should be what kind of geometrical object? A Circle B Cylinder C Cube Answer D Sphere E Rectangle

Sphere Return to Table of Contents

Symmetric Charge Distributions There are three common symmetric charge distributions that are used to illustrate Gauss's Law. They are: Sphere (both conducting and insulated) Line of charge (conducting and insulated) Infinite plane of charge We'll start with the Sphere.

Insulated Sphere An insulated sphere of radius, R, is uniformly positively charged with a volume charge density, ρ and total charge Q. Find the Electric Field inside and outside the sphere. Q R r Start with the field outside the sphere. Construct a spherical Gaussian surface of radius, r. The Electric Field is perpendicular and constant everywhere on the surface, so Gauss's Law is appropriate.

Insulated Sphere Q R r The closed surface integral of the sphere is the equation for the surface area of a sphere!

Insulated Sphere Calculate the Electric Field within the sphere. Construct a Gaussian surface within the sphere. Only the enclosed charge contributes to the field, so Q can't be used. Q r R We have to find the part of Q that exists within the Gaussian surface. Assume a uniformly distributed charge and take the ratio of the two spheres times the total charge:

Insulated Sphere Q R r We have a two part solution for the Insulated Sphere's Electric Field: for r > R for r < R

Insulated Sphere Q R for r > R r for r < R For r = R, both expressions give the same value for the Electric Field. That's good. For r > R, the Electric Field due to an insulated sphere is the same as a point charge located at the center of the sphere. A sphere acts as a point charge with the field measured from the center of the sphere.

Conducting Sphere or Thin Shell of Charge Q Q R R r r For both of these cases, the Electric Field outside is the same as for the Insulated sphere, as the enclosed charge is Q.

Conducting Sphere or Thin Shell of Charge Q Q R R r r However, since the charge for a conducting sphere resides on the surface, when we draw a Gaussian surface right inside the sphere, we get an enclosed charge of zero - just like a surface within the thin shell, hence, zero Electric Field for both cases.

Conducting Sphere or Thin Shell of Charge Q Q R R r r Again, there is a two part solution for this case. for r > R for r < R

8 A metal sphere of radius R is given a positive charge, q. What is the electric field at a distance r, where r > R? A B C Answer D E

9 A metal sphere of radius R is given a positive charge, q. What is the electric field at a distance r, where r < R? A B C Answer D E

10 A metal sphere of radius R is given a positive charge, q. What is the electric field at a distance r, where r = R? A B C Answer D E

11 Which is the graph of the Electric Field for a charged conducting sphere of radius R? A B C E E max E max E max E E Answer R R R D E E E E max E max R R

12 An insulated sphere of radius R has a uniform charge distribution of magnitude Q. Which is the graph of its Electric Field? A B C E E E E max E max E max Answer R R R D E E E E max E max R R

13 A nonconducting sphere of radius R has a uniform charge distribution of magnitude Q. What is the electric field at a distance less then R? A B C D E R Answer

Infinite Rod of Charge Return to Table of Contents

Infinite Rod of Charge Consider an infinite rod of charge of linear density λ and radius R. Find the Electric field inside and outside the rod. Construct a Gaussian surface (in blue-green) taking advantage of the cylindrical symmetry of the wire - at a distance r from the wire, the Electric Field is constant and perpendicular to the surface of the cylinder. Since the wire is infinite, we don't have to worry about the ends of the wire - where there would be a parallel component of the Electric Field and Gauss's Law can't be applied.

Infinite Rod of Charge Surface area of the curved part of the cylinder This works whether the rod is insulated or a conductor. Next, find the Electric Field within the rod.

Infinite Rod of Charge If the rod is a conductor, then the Electric Field within the rod is zero - there is no enclosed charge. For an insulator of volume charge density ρ, we'll construct a Gaussian surface of length l and radius r within the rod. Since the rod is of infinite length, there is no Electric Field through the end caps of the Gaussian cylinder. The Electric Field is uniform and perpendicular to the curved surface.

Infinite Rod of Charge Enclosed charge is the volumetric charge density times the volume of the cylinder. Surface area of the cylinder.

14 There is an electric charge distributed uniformly over an infinitely long metal wire of radius R. What is the electric field at a distance r (r > R) from the center of the wire?(the charge per unit of length is.) A B C Gaussian Surface Answer D E

15 There is an electric charge distributed uniformly over an infinitely long metal wire of radius R. What is the electric field at a distance r (r < R) from the center of the wire?(the volume charge density is.) A B C Gaussian Surface Answer D E

Infinite Plane of Charge Return to Table of Contents

Infinite Plane of Charge Here is an infinite plane (pretend the plane extends infinitely in the x and y direction) of positive charge with an Electric Field pointing out of the page. Find the magnitude of the field. The surface charge density on the plane is σ. Rotate the plane sideways so it looks like an infinite line.

Infinite Plane of Charge Draw a Gaussian surface - a cylinder - which is composed of two end caps, each with a surface area of A, and a curved surface (open tube). End Cap End Cap Curved surface

Every point on the curved surface is perpendicular to the Electric field generated by the positively charged plane of charge. Infinite Plane of Charge The curved surface does not contribute anything to the net Electric Field. End Cap End Cap Curved surface

Infinite Plane of Charge There are two end caps, each with the Electric Field perpendicular to the surface area of the cap: The flux through one cap is. Since there are two caps, the End Cap total flux through the Gaussian surface of the two caps and the curved surface is. End Cap Curved surface

Infinite Plane of Charge The charge enclosed by the Gaussian surface is equal to the area of the infinite plane that is subtended by the end caps, A, times the surface charge density, σ. Putting this together with the flux determined on the previous page: End Cap End Cap Curved surface The Electric Field is constant near the infinite plane.

Symmetry Summary Gauss's Law was used to solve for three symmetries, where R is the radius of the sphere or rod. Spherical Rod for r > R for r < R for r > R These solutions are for uniform distributions of charge (insulators). If the sphere or line of charge is a conductor than E = 0 for r < R. Infinite plane for r < R

16 Describe the Electric Field dependence on r as Gauss's Law is used to solve for a sphere, a rod and an infinite plane of charge. Answer

17 What is the magnitude of the Electric Field right outside an infinite plane of charge? A B C Answer D E

18 What is the shape and orientation of the Gaussian surface used to find the Electric Field outside of an infinite plane of charge? A A sphere through the surface. B A cube parallel to the surface. C A cube through the surface. Answer D A cylinder parallel to the surface. E A cylinder perpendicular to the surface.

Electrostatic Equilibrium of Conductors Return to Table of Contents

Electrostatic Equilibrium of Conductors Charges are free to move within the body of a conductor - and there are a great number of these charges. Electrostatic Equilibrium occurs when there is no net movement of these free charges. A conductor in electrostatic equilibrium has the following properties: The Electric Field within the conductor is equal to zero. Any excess charge on an isolated conductor (non grounded) resides on its surface. The Electric Field outside the conductor is perpendicular to the surface and equal to the charge density divided by the electrical permittivity.

Zero Electric Field The Electric Field within the conductor is equal to zero. If there were an Electric Field within the conductor, then there would be a force on the charges, which would result in a net movement of the charges - hence, no equilibrium. But what if an external Electric Field is applied to the conductor??

Zero Electric Field The charges will rearrange themselves - the negative charges will be attracted to the left side of the conductor, leaving an equal positive charge on the right side. This creates an opposing field to the applied field. The charges will move until this field is equal and opposite - then the charges will once again be in electrostatic equilibrium. This process takes nanoseconds. - - - - - - -

Excess Charge on Surface Gauss's Law is used to show that any excess charge on a conductor must reside on its surface. Create a Gaussian surface just inside any arbitrarily shaped conductor. The Electric Field inside a conductor in electrostatic equilibrium is zero. By Gauss's Law, if the Electric Field is zero, then the net enclosed charge is zero. The Gaussian surface can be drawn infinitely close to the surface of the conductor - thus any free charge must reside on the surface of the conductor.

External Field perpendicular The external Electric Field is perpendicular to the surface. Construct a Gaussian cylinder that pokes through the surface. We'll reuse the cylinder that was used for an infinite plane of charge. If the cylinder is small enough, the surface looks flat. End Cap End Cap Curved surface Unlike the infinite plane, there is no Electric Field pointing to the left of the Gaussian surface because the Electric Field equals zero within a conductor.

External Field perpendicular There is no Electric Field parallel to the surface - if there was, a force would be exerted on the charges and they would move - they would not be in electrostatic equilibrium. Thus, the Electric Field is perpendicular and pointing outside the surface at every point. End Cap End Cap Curved surface

Charge Distribution within a Conductor Take a conductor with charge Q. The entire charge resides on the surface as shown below. Now punch a hole through the conductor, creating an open cavity. The charge still is on the outside surface as shown Q by the blue Gaussian surface drawn within the conductor - there is no Electric Field within the conductor, hence no enclosed charge. Q

Charge Distribution within a Conductor Place a charge, Q within the cavity. What happens to the charge distribution? - - - -Q Q - - - This charge will generate an electric field. But, an electric field cannot exist within a conductor at electrostatic equilibrium. The charges within the conductor will redistribute themselves such that there will be an enclosed charge of zero within the Gaussian surface. This results in an induced charge of -Q on the inner surface of the conductor. But where did this come from?

Charge Distribution within a Conductor - - - -Q Q - - - 2Q Conservation of Charge requires that the conductor maintains its total original charge of Q. Since -Q is on the inside, 2Q will be distributed on the external surface - summing to Q for the conductor.

19 An isolated conductor is placed within an external electric field that creates an electric field within the conductor. What order of magnitude of time will it take for polarization to occur within the conductor to neutralize this field? A 10-12 s B 10-9 s C 10-6 s Answer D 10-3 s E 10-1 s

20 What is the magnitude of the Electric Field right outside an arbitrarily sized conductor in electrostatic equilibrium? A B C D Answer E

21 Which of these are properties of a conductor in electrostatic equilibrium? A I B II C III D I and II I - The Electric Field is a constant, non zero value within the conductor. II - Excess charge resides on the surface of the conductor. III - The Electric field just outside the conductor is perpendicular to the surface. Answer E II and III

22 A conductor with an outer surface B and an inner surface A surrounding a hole is charged to 2Q. A charge of -3Q is placed within the hole. What is the resulting charge distribution on the conductor? B B A A -3Q 2Q A There is no change since the 2Q is on surface B. B A charge of -Q is uniformly distributed throughout the conductor. C Surface A is charged to -3Q and surface B is charged to 2Q. D Surface A is charged to 3Q and surface B is charged to -Q. E Surface A is charged to -2Q and surface B is charged to 2Q. Answer