Lecture 6.2

Lecture 6.2 Chapter 30. Potential and Field Chapter Goal: Tounderstand how the electric potential is connected to the electric field.

Review of Last Chapter • Electrostatic potential energy and energy conservation • Electrostatic potential • How to calculate potential by a charge distribution • Relation between potential and electric field

Chapter 30. Potential and Field Topics of this lecture: • Relation between potential and field • Revisit Conductors in Electrostatic Equilibrium • Sources of Electric Potential: battery

Relation between potential and field Consider a test charge q in an electric field, its potential energy is q V(r). If we move q slowly from ri to rf , the work done by the field is negative the change of potential energy: Therefore we have Thus we can find the potential difference between two points if we know the electric field.

Equipotential Surface

What we have learned about conductors: • Vanishign field inside conductors • No net charge inside • Field perpendicular to the surface right outside • Field proportional to surface charge density Revisit Conductors in Equilibrium • What we can learn now: • Potential everywhere constant inside conductors • The surface of a conductor is an equipotential surface

Two Charged Conductor Spheres Connected by Wire

A Cartoon for Conductor

Finding the Electric Field from the Potential

Electrostatic force/field is conservative The work done by the electric field is the same whether the test charge follows path 1 or path 2. If the charge q follows a close loop and come back to the initial position, the total work done by the electrostatic force is zero. The electrostatic force/field is conservative, just like the gravitational force/field.

Kirchhoff’s Loop Law For any path that starts and ends at the same point Stated in words, the sum of all the potential differences encountered while moving around a loop or closed path is zero. This statement is known as Kirchhoff’s loop law.

How a Van de Graph Works

How a Battery Works

Batteries and emf The potential difference between the terminals of an ideal battery is In other words, a battery constructed to have an emf of 1.5V creates a 1.5 V potential difference between its positive and negative terminals. The total potential difference of batteries in series is simply the sum of their individual terminal voltages:

Lecture 7.1 Announcement: One home work with lowest score will be dropped at the end of semester. The second midterm in the next Thursday Some students complained that it was noisy in the back. Please keep quiet in the room, and go outside if you really need to talk. Today’s office hour ends at 2:30pm.

Review: Potential and Field • Electric potential energy and electric potential • Relation between potential and field • Revisit Conductors in Electrostatic Equilibrium • Sources of Electric Potential: battery

What total potential difference is created by these three batteries? • 1.0 V • 2.0 V • 5.0 V • 6.0 V • 7.0 V

Which potential-energy graph describes this electric field?

Three charged, metal spheres of different radii are connected by a thin metal wire. The potential and electric field at the surface of each sphere are V and E. Which of the following is true? • V1 = V2 = V3 and E1 > E2 > E3 • V1 > V2 > V3 and E1 = E2 = E3 • V1 = V2 = V3 and E1 = E2 = E3 • V1 > V2 > V3 and E1 > E2 > E3 • V3 > V2 > V1 and E1 = E2 = E3

This lecture • Capacitance and Capacitors • The Energy Stored in a Capacitor • Dielectrics

Capacitor: two piece of oppositely charged conductors

Cylindrical and Spherical Capacitors

More Capacitor

Capacitor: what is it good for?

Capacitance and Capacitors The ratio of the charge Q to the potential difference ΔVC is called the capacitance C: Capacitance is a purely geometric property of two electrodes because it depends only on their surface area and spacing. The SI unit of capacitance is the farad: 1 farad = 1 F = 1 C/V. The charge on the capacitor plates is directly proportional to the potential difference between the plates. Result of principle of linear superposition

Parallel Plate Capacitor

Combinations of Capacitors

Combinations of Capacitors If capacitors C1, C2, C3, … are in parallel, their equivalent capacitance is If capacitors C1, C2, C3, … are in series, their equivalent capacitance is

Rank in order, from largest to smallest, the equivalent capacitance (Ceq)a to (Ceq)d of circuits a to d. • (Ceq)d > (Ceq)b > (Ceq)a > (Ceq)c • (Ceq)d > (Ceq)b = (Ceq)c > (Ceq)a • (Ceq)a > (Ceq)b = (Ceq)c > (Ceq)d • (Ceq)b > (Ceq)a = (Ceq)d > (Ceq)c • (Ceq)c > (Ceq)a = (Ceq)d > (Ceq)b

The Energy Stored in a Capacitor • Consider charging a capacitor starting from zero charge: • At the initial state, the energy stored is zero • At an intermediate stage with charge q, we add dq more charge onto the capacitor, the work that we need to do is

Where is the electric energy?

The Energy in the Electric Field The energy density of an electric field, such as the one inside a capacitor, is The energy density has units J/m3. Electromagnetic field carries energy

Dielectrics • The dielectric constant, like density or specific heat, is a property of a material. • Easily polarized materials have larger dielectric constants than materials not easily polarized. • Vacuum has κ = 1 exactly.

Inserting a dielectrics into a capaticor

Dielectrics Changes Capacitance

Dielectrics • Filling a capacitor with a dielectric increases the capacitance by a factor equal to the dielectric constant.

Chapter 30. Summary Slides

General Principles

Important Concepts

Applications

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