PHYS 241 Exam 1 Review

1. PHYS 241 Exam 1 Review Kevin Ralphs

2. Overview • General Exam Strategies • Concepts • Practice Problems

3. General Exam Strategies • Don’t panic!!! • If you are stuck, move on to a different problem to build confidence and momentum • Begin by drawing free body diagrams • “Play” around with the problem • Take fifteen to twenty minutes before the exam to relax… no studying. • Look for symmetries

4. Concepts • Electrostatics • Coulomb’s Law • Principle of Superposition • Electric Field • Continuous Charge Distributions • Conductors vs. Insulators • Gauss’s Law • Potential • Capacitance

5. Electrostatics • Our study of electric fields so far has been based on a few assumptions • These assumptions collectively are known as the electrostatic approximation • Basically we assume that our systems have to come to a dynamic equilibrium before we do our calculations • We will be ignoring transitory behavior or steady state behaviors (no currents or magnetic fields)

6. Coulomb’s Law • What does it tell me? • It tells you the force between two charged particles • Why do I care? • Forces describe the acceleration a body undergoes • The actual path the body takes in time can be found from the acceleration in two ways • Use integration to get the particle’s velocity as a function of time, then integrate again to gets its position • Kinematic equations (the result when method 1. is applied in the case of constant acceleration)

7. Coulomb’s Law • Forces have magnitude and direction so Coulomb’s law tells you both of these • Magnitude: • Direction: Along the line connecting the two bodies. It is repulsive in the case of like charges, attractive for opposite charges

8. Principle of Superposition • What does it tell me? • The electric force between two bodies only depends on the information about those two bodies • Why do I care? • Essentially, all other charges can be ignored, the result obtained in pieces and then summed… this is much simpler

9. Electric Field • What does it tell me? • A vector proportional to the force a positive test charge would experience at a point in space • Why do I care? • Calculating the force a particular charge feels doesn’t directly tell you how other charges would behave • The electric field gives you a solution that applies to any charge, so it reduces your work

10. Electric Field • Electric field due to a point charge at distance r with charge q • Principle of superposition still applies • You can sum individual fields due to discrete charges • You can integrate continuous charge distributions where the charge becomes and the field becomes

11. Continuous Charge Distributions • Motivation for the equation: • Very far from a charge distribution, it looks like a point charge • So if we “chop” up the distribution into small enough pieces, each one will have a field contribution we can calculate • The principle of superposition then allows the integrand to approach the true field

12. Continuous Charge Distributions • General procedure to setup the integrals • Prepare your integral • Change integral to integrate over where the charge lies (aka parameterization) • Identify elements of the integrand that depend on the integrating variable • Determine explicit relationships with the integrating variable • Integrate

13. Conductors vs Insulators • Conductors • All charge resides on the surface, spread out to reduce the energy of the configuration • The electric field inside is zero • The potential on a conductor is constant (i.e. the conductor is an equipotential) • The electric field near the surface is perpendicular to the surface Note: These are all logically equivalent statements

14. Conductors vs Insulators • Insulators • Charge may reside anywhere within the volume or on the surface and it will not move • Electric fields are often non-zero inside so the potential is changing throughout • Electric fields can make any angle with the surface

15. Gauss’s Law • What does it tell me? • The electric flux (flow) through a closed surface is proportional to the enclosed charge • Why do I care? • You can use this to determine the magnitude of the electric field in highly symmetric instances • Flux through a closed surface and enclosed charge are easily exchanged

16. 3 Considerations for Gaussian Surfaces Gauss’s law is true for any imaginary, closed surface and any charge distribution no matter how bizarre. It may not be useful, however. • The point you are evaluating the electric field at needs to be on your surface • Choose a surface that cuts perpendicularly to the electric field (i.e. an equipotential surface) • Choose a surface where the field is constant on the surface *Note this requires an idea of what the field should look like

17. Common Gauss’s Law Pitfalls • Your surface must be closed • The charge you use in the formula is the charge enclosed by your surface • The Gaussian surface need not be a physical surface • Start from the definition of flux and simplify only if your surface allows it

18. Potential • What does it tell me? • The change in potential energy per unit charge an object has when moved between two points • Why do I care? • The energy in a system is preserved unless there is some kind of dissipative force • So the potential allows you to use all the conservation of energy tools from previous courses (i.e. quick path to getting the velocity of a particle after it has moved through a potential difference)

19. Potential • Why do I care? (cont.) • If you have the potential defined over a small area, the potential function encodes the information about the electric field in the derivative

20. Potential • Word of caution: • Potential is not the same as potential energy, but they are intimately related • Electrostatic potential energy is not the same as potential energy of a particle. The former is the work to construct the entire configuration, while the later is the work required to bring that one particle in from infinity • There is no physical meaning to a potential, only difference in potential matter. This means that you can assign any point as a reference point for the potential

21. Capacitance • What does it tell me? • The charge that accumulates on two conductors is proportional to the voltage between them • Why do I care? • Capacitors are vital components in electronics • They can be used to temporarily store charge and energy, and play an even more important role when we move to alternating current systems • Camera flashes, touch screen devices, modern keyboards all exploit capacitance

22. Capacitance • In circuits • In well-behaved configurations, capacitors may be combined into a single equivalent capacitor • Parallel * This is like increasing the area of the plates * • Series * This is like increasing the separation distance *

23. Capacitance • Dielectric • Put simply, a dielectric is a material (an insulator) that weakens the electric field around it • This allows more charge to be placed on the plates for the same voltage (i.e. capacitance is increased) • The permittivity of a dielectric tells you how it affects the capacitance • The ratio of the permittivity of a dielectric and the permittivity of free space is the dielectric constant

24. Capacitance • Capacitors are in equilibrium… • Series: when they have the same charge • Parallel: when they have the same voltage

25. Practice Problems

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