2.5 Conductors

1. 2.5 Conductors 2.5.1 Basic Properties of Conductors 2.5.2 Induced Charges 2.5.3 The Surface Charge on a Conductor; the Force on a Surface Charge 2.5.4 Capacitors

2. are free to move in a conductor (1) inside a conductor otherwise, the free charges that produce will move to make inside a conductor 2.5.1 Basic Properties of Conductors (2) ρ = 0 inside a conductor (3) Any net charge resides on the surface (4) V is constant, throughout a conductor.

3. uniform uniform 2.5.1 (2) (5) is perpendicular to the surface, just outside a conductor. Otherwise, will move the free charge to make in terms of energy, free charges staying on the surface have a minimum energy.

4. 2.5.1 (3) Example : A point charge q at the center of a spherical conducting shell. How much induced charge will accumulate there? Solution : q induced charge conservation

5. a Faraday cage can shield out stray 2.5.2 Induced Charge Example 2.9 Within the cavity is a charge +q. What is the field outside the sphere? -q distributes to shield q and to make i.e., from charge conservation and symmetry. +q uniformly distributes at the surface a,b are arbitrary chosen

6. 2.5.3 The Surface Charge on a Conductor or if we know V, we can get s.

7. 2.5.3 electrostatic pressure Force on a surface charge, why the average? In case of a conductor

8. 2.5.4 Capacitors Consider 2 conductors (Fig 2.53) The potential difference (V is constant.) Define the ratio between Q and V to be capacitance a geometrical quantity in mks 1 farad(F)= 1 Coulomb / volt inconveniently large ;

9. 2.5.4 (2) +Q -Q Example 2.10 Find the capacitance of a “parallel-plate capacitor”? Solution:

10. 2.5.4(3) Example 2.11 Find capacitance of two concentric spherical shells with radii a and b . -Q +Q Solution:

11. 2.5.4(4) The work to charge up a capacitor

12. Energy Storage in Capacitors, Dielectrics and Capacitors With Dielectrics • Today’s agenda: • Energy Storage in Capacitors. • You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered. • Dielectrics. • You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

13. Energy Storage in Capacitors Let’s calculate how much work it takes to charge a capacitor. The work required for an external force to move a charge dq through a potential difference V is dW = dq V. From Q=CV ( V = q/C): V + - dq q is the amount of charge on the capacitor during the time the charge dq is being moved. + We start with zero charge on the capacitor, and end up with Q, so +q -q

14. The work required to charge the capacitor is the amount of energy you get back when you discharge the capacitor (because the electric force is conservative). Thus, the work required to charge the capacitor is equal to the potential energy stored in the capacitor. Because C, Q, and V are related through Q=CV, there are three equivalent ways to write the potential energy.

15. All three equations are valid; use the one most convenient for the problem at hand. It is no accident that we use the symbol U for the energy stored in a capacitor. It is just another “version” of electrical potential energy. You can use it in your energy conservation equations just like any other form of potential energy!

16. Example: a camera flash unit stores energy in a 150 F capacitor at 200 V. How much electric energy can be stored? If you keep everything in SI (mks) units, the result is “automatically” in SI units.

17. Energy Stored in Electric Fields Energy is stored in the capacitor: V + - E d +Q -Q area A The “volume of the capacitor” is Volume=Ad

18. Energy stored per unit volume (u): V + - The energy is “stored” in the electric field! E We’ve gone from the concrete (electric charges experience forces)… d +Q -Q …to the abstract (electric charges create electric fields)… area A …to an application of the abstraction (electric field contains energy).

19. “The energy in electromagnetic phenomena is the same as mechanical energy. The only question is, ‘Where does it reside?’ In the old theories, it resides in electrified bodies. In our theory, it resides in the electromagnetic field, in the space surrounding the electrified bodies.”—James Maxwell V + - E This is not a new “kind” of energy. It’s the electric potential energy resulting from the coulomb force between charged particles. f +Q -Q Or you can think of it as the electric energy due to the field created by the charges. Same thing. area A

20. Today’s agenda: • Energy Storage in Capacitors. • You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered. • Dielectrics. • You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

21. Dielectrics If an insulating sheet (“dielectric”) is placed between the plates of a capacitor, the capacitance increases by a factor , which depends on the material in the sheet.  is the dielectric constant of the material. dielectric In general, C = 0A / d.  is 1 for a vacuum, and  1 for air. (You can also define  = 0 and write C =  A / d).

22. The dielectric is the thin insulating sheet in between the plates of a capacitor. dielectric Any reasons to use a dielectric in a capacitor? Makes your life as a physics student more complicated. Lets you apply higher voltages (so more charge). Lets you place the plates closer together (make d smaller). Increases the value of C because >1. Gives you a bigger kick when you discharge the capacitor through your tongue! Gives you a bigger kick when you discharge the capacitor through your tongue! Gives you a bigger kick when you discharge the capacitor through your tongue!

23. 2.3 Electric potential 2.3.1 Introduction to Potential 2.3.2 Comments on Potential 2.3.3 Poisson’s Equation and Laplace’s Equation 2.3.4 The Potential of a Localized Charge Distribution 2.3.5 Electrostatic Boundary Conditions

24. 2.3.1 introduction to potential Any vector whose curl is zero is equal to the gradient of some scalar. We define a function: Where is some standard reference point ; V depends only on the point P. V is called the electric potential. The fundamental theorem for gradients so

25. 2.3.2 Comments on potential (1)The name Potential is not potential energy V : Joule/coulomb U : Joule

26. 2.3.2 (2) are not independent functions 2 (2)Advantage of the potential formulation V is a scalar function, but E is a vector quantity If you know V, you can easily get E: 1 3 so 1 2 3

27. 2.3.2 (3) (3)The reference point Changing the reference point amounts to adds a constant to the potential (Where K is a constant) Adding a constant to V will not affect the potential difference between two point: Since the derivative of a constant is zero: For the different V, the field E remains the same. Ordinarily we set

28. 2.3.2 (4) (4)Potential obeys the superposition principle , Dividing through by Q Integrating from the common reference point to p , (5)Unit of potential Volt=Joule/Coulomb Joule/Coulomb

29. 2.3.2 (5) Example 2.6 Find the potential inside and outside a spherical shell of radius R, which carries a uniform surface charge (the total charge is q). solution: for r>R: _ for r<R:

30. 2.3.3 Poisson’s Eq. & Laplace’s Eq. Poisson’s Eq. Laplace’s eq.

31. 2.3.4 The Potential of a Localized Charge Distribution R • Potential for a point charge • Potential for a collection of charge Ri

32. 2.3.4 (2) • Potential of a continuous distribution for volume charge for a line charge for a surface charge • Corresponding electric field [ ]

33. 2.3.4 (3) Example 2.7 Find the potential of a uniformly charged spherical shell of radius R. Solution: r

34. 2.3.4 (4)

35. Superposition • Coulomb law 2.3.5 Electrostatic Boundary Condition Electrostatic problem The above equations are differential or integral. For a unique solution, we need boundary conditions. (e.q. , V()=0 ) (boundary value problem. Dynamics: initial value problem.)