APSC 178 Chapter 2 - Lecture notes 2 PDF

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UBC School of Engineering

Prof. Kenneth Chau

APSC 178: Electricity and Magnetism Chapter 2: Gauss’s Law In this Chapter, we will learn about the basic concepts behind Gauss’s law. Here is the web of concepts we will be developing in this Chapter:

Some questions that will be answered in this Chapter include: -

How can you determine the amount of charge within a closed surface by examining the electric field on the surface? What is electric flux? How do you calculate electric flux? What is Gauss’ Law? How do we use it? Where is charge located on a conductor? Why?

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UBC School of Engineering

Prof. Kenneth Chau

The Relationship Between Charge and Flux In the last Chapter, we asked how to find an electric field (at a given point) produced by some charge distribution. We found the field by breaking the charge distribution down to an assembly of small point charges, then adding everything up. There is a more elegant way to find the electric field due to a charge distribution by examining the relationship between charge and electric flux. Let’s try the following thought experiment:

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UBC School of Engineering

Prof. Kenneth Chau

What did you notice about these thought experiments?

CP: A certain region of space bounded by an imaginary closed surface contains no charge. Is the electric field always zero everywhere on the surface? Why or why not?

CP: A spherical surface encloses a point charge q. If the point charge is moved from the centre of the sphere closer to the edge, does the electric flux at a given point on the sphere change? Does the total flux through the surface charge?

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UBC School of Engineering

Prof. Kenneth Chau

Electric Flux In the last section, we qualitatively discussed the relationship between the amount of field lines crossing an area (electric flux) and charge. Now, let’s try to calculate the electric flux mathematically. Let’s take baby steps. We’ll start by considering the electric flux for a uniform electric field through a planar loop of fixed area A.

Notice that in the case of an open surface, there are two choices for the normal unit vector. For a closed surface, we define the normal unit vector to be positive when facing outward.

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UBC School of Engineering

Prof. Kenneth Chau

For the more general case where the electric field is non-uniform or the surface is non-planar, we will generally define the electric flux through calculus (remember, calculus is just the math of breaking things down into tiny pieces).

AP: A point charge q is surrounded by a sphere with radius r. Find the electric flux through the sphere due to this charge.

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UBC School of Engineering

Prof. Kenneth Chau

Gauss’ Law The total electric flux through any closed surface is proportional to the total (net) electric charge enclosed by the surface. Note that the “surface” is an imaginary construct and is not composed of real materials. In mathematical notation, Gauss’ Law is given by:

Gauss’ Law is easy to state, but more challenging to apply. The trick with Gauss’ Law is that we wouldn’t ask you to solve it on pen and paper unless it was doable. The first thing you should notice is that we only use it when there is plenty of symmetry. In the next few examples, I will show you progressively more complex applications of Gauss’ Law, starting with uniform charge distributions with spherical, cylindrical, and planar symmetries, and then an example where the charge distribution is non-uniform. You will hopefully notice a pattern emerging.

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UBC School of Engineering

Prof. Kenneth Chau

AP: A positive charge q is uniformly distributed over a shell of radius R. What is the symmetry of this geometry? Find the electric field inside and outside the shell by applying Gauss’ Law. What is an appropriate Gaussian surface?

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UBC School of Engineering

Prof. Kenneth Chau

AP: Electric charge is distributed uniformly along an infinitely long, thin wire. The charge per unit length is λ (positive). What is the symmetry of this geometry? Find the electric field by applying Gauss’ Law. What is an appropriate Gaussian surface?

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UBC School of Engineering

Prof. Kenneth Chau

AP: Consider a thin, flat, infinite sheet on which there is uniform charge per unit area σ. What is the symmetry of this geometry? Find the electric field by applying Gauss’ Law. What is an appropriate Gaussian surface?

CP: What if we had two infinite parallel sheets of charges of equal magnitude and opposite sign; one with a charge density +σ and the other with a charge density -σ. Without any math, what would be the electric field between the plates? Outside?

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UBC School of Engineering

Prof. Kenneth Chau

AP: A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ(r) is given by (r )  , r  R / 2  (r )  2 (1 r / R ), R / 2  r  R  (r )  0, r  R

(a) Find α in terms of Q and R. (b) Using Gauss’ law, derive an expression for the magnitude of the electric field as a function of r. Do this separately in all three regions. Express your answer in terms of the total charge Q. (c) What fraction of the total charge is in the region r ≤ R/2?

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UBC School of Engineering

Prof. Kenneth Chau

Charges on Conductors Let’s say you place some charge on a conductor. Where would you expect that charge to go? Why?

If you believe that charge should only exist on the surface and the field everywhere inside a conductor must be zero (or else you would have perpetual current), then let’s look at what happens if we place a cavity inside a conductor. This thought experiment will reveal how charges are induced on the interior and exterior surfaces of the conductor.

CP: Let say you have a conductor with a cavity. In the cavity, you place a positive charge +q and a negative charge –q. What will be the induced charge on the inner and outer walls of the conductor?

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UBC School of Engineering

Prof. Kenneth Chau

Fields at the Surface of a Conductor So we now know that no fields can exist inside a conductor. What happens to the fields at the surface of a conductor? How do you expect that they will be oriented? Why? Try writing out your guess here first and include an argument as to why you believe so:

Now, I’ll go through an argument using Gauss’ Law:

Compare your guess to the argument using Gauss’ Law. Any similarities? Differences?

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UBC School of Engineering

Prof. Kenneth Chau

AP: The earth (a conductor) has a net electric charge. The resulting electric field near the surface can be measured and its average value is 150N/C, directed toward the centre of the earth. (a) What is the corresponding surface charge density? (b) What is the total surface charge of the earth?

Summary An idea web of the concepts explored in this Chapter and how they relate to the overall course.

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