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Ben Gurion University of the Negev www.bgu.ac.il/atomchip Physics 2B for Materials and Structural Engineering Lecturer: Daniel Rohrlich Teaching Assistants: Oren Rosenblatt, Shai Inbar Week 4. Potential, capacitance and capacitors – E from V • equipotential surfaces • E in and on a conductor • capacitors and capacitance • capacitors in series and in parallel Source: Halliday, Resnick and Krane, 5th Edition, Chaps. 28, 30.

With 100,000 V on her body, why is this girl smiling???

E from V Let U(r2) be the potential energy of a charge q at the point r2. We found that it is minus the work done by the electric force Fq in bringing the charge to r2: If we divide both sides by q, we get (on the left side) potential instead of potential energy, and (on the right side) the electric field instead of the electric force:

E from V Let V(r2) be the electric potential at the point r2. We have just obtained it from the electric field E: Now let’s see how to obtain E from V. For r2 close to r1 we can write r2 = r1 + Δr and expand V(r2) in a Taylor series:

E from V We can write this expansion more compactly using the “del” (gradient) operator: so with all the derivatives evaluated at the point r1. We also write

E from V So now we can write and therefore

E from V The equation is a scalar equation, but since we can vary each component of Δr independently, it actually yields three equations:

E from V The equation is a scalar equation, but since we can vary each component of Δr independently, it actually yields three equations, i.e.

E from V which reduce to a single vector equation:

E from V Example 1 (Halliday, Resnick and Krane, 5th Edition, Chap. 28, Exercise 34): Rutherford discovered, 99 years ago, that an atom has a positive nucleus with a radius about 105 times smaller than the radius R of the atom. He modeled the electric potential inside the atom (r < R) as follows: where Z is the atomic number (number of protons). What is the corresponding electric field?

E from V Answer: since

E from V Example 2: Electric field of a dipole We found that the electric potential V(x,y,z) of a dipole made of charges q at (0,0,d/2) and –q at (0,0,–d/2) is z d/2 r– (x,y,z) r+ –d/2

E from V z d/2 r– (x,y,z) r+ –d/2

E from V Now we will consider the case d << r–, r+ and use these rules for 0 < α << 1 (derived from Taylor or binomial expansions) to approximate E:

E from V For d << r–, r+ we have z r– d/2 (x,y,z) r+ –d/2

E from V So z r– d/2 (x,y,z) r+ –d/2

E from V We can check that these results coincide with the results we obtained for the special cases x = 0 = y and z = 0: z r– d/2 (x,y,z) r+ –d/2

E from V This may look complicated but it is easier than calculating Ex, Ey and Ez directly! z r– d/2 (x,y,z) r+ –d/2

Equipotential surfaces We have already seen equipotential surfaces in pictures of lines of force: and in connection with potential energy: r2 r2 F(r) r r1 r1

Equipotential surfaces All points on an equipotential surface are at the same electric potential. Electric field lines and equipotential surfaces meet at right angles. Why? r2 r2 F(r) r r1 r1

Equipotential surfaces The surface of a conductor at electrostatic equilibrium – when all the charges in the conductor are at rest – is an equipotential surface, even if the conductor is charged: Small pieces of thread (in oil) align with the electric field due to two conductors, one pointed and one flat, carrying opposite charges. [From Halliday, Resnick and Krane]

Equipotential surfaces We can also visualize the topography of the electric potential from the side (left) and from above (right) with equipotentials as horizontal curves. V y x

Equipotential surfaces Quick quiz: In three space dimensions, rank the potential differences V(A) – V(B), V(B) – V(C), V(C) – V(D) and V(D) – V(E). B A 9 V E 8 V D C 7 V 6 V

E in and on a conductor Four rules about conductors at electrostatic equilibrium: 1. The electric field is zero everywhere inside a conductor. Explanation: A conductor contains free charges (electrons) that move in response to an electric field; at electrostatic equilibrium, then, the electric field inside a conductor must vanish. Example: An infinite conducting sheet in a constant electric field develops surface charges that cancel the electric field inside the conductor. - - - - - - - + + + + + + +

E in and on a conductor Four rules about conductors at electrostatic equilibrium: 1. The electric field is zero everywhere inside a conductor. 2. Any net charge on an isolated conductor lies on its surface.

E in and on a conductor Four rules about conductors at electrostatic equilibrium: 1. The electric field is zero everywhere inside a conductor. 2. Any net charge on an isolated conductor lies on its surface. Explanation: A charge inside the conductor would imply, via Gauss’s law, that the electric field could not be zero everywhere on a small surface enclosing it, contradicting 1.

E in and on a conductor Four rules about conductors at electrostatic equilibrium: 1. The electric field is zero everywhere inside a conductor. 2. Any net charge on an isolated conductor lies on its surface. 3. The electric field on the surface of a conductor must be normal to the surface and equal to σ/ε0, where σ is the surface charge density.

E in and on a conductor Four rules about conductors at electrostatic equilibrium: 1. The electric field is zero everywhere inside a conductor. 2. Any net charge on an isolated conductor lies on its surface. 3. The electric field on the surface of a conductor must be normal to the surface and equal to σ/ε0, where σ is the surface charge density. Explanation: A component of E parallel to the surface would move free charges around. (Hence the surface of a conductor at electrostatic equilibrium is an equipotential surface, even if the conductor is charged.)

E in and on a conductor Let’s compare the electric fields due to two identical surface charge densities σ, one on a conductor with all the excess charge on one side (e.g. the outside of a charged sphere) and the other on a thin insulating sheet. The figure shows a short “Gaussian can” straddling the thin charged sheet. If the can is short, we need to consider only the electric flux through the top and bottom. Gauss’s law gives 2EA = ФE = σA/ε0, so E = σ/2ε0. But if there is flux only through the top of the can, Gauss’s law gives EA = ФE = σA/ε0 and E = σ/ε0. “Gaussian can”

E in and on a conductor Four rules about conductors at electrostatic equilibrium: 1. The electric field is zero everywhere inside a conductor. 2. Any net charge on an isolated conductor lies on its surface. 3. The electric field on the surface of a conductor must be normal to the surface and equal to σ/ε0, where σ is the surface charge density. 4. On an irregularly shaped conductor, the surface charge density is largest where the curvature of the surface is largest.

E in and on a conductor Explanation: If we integrate E·dr along any electric field line, starting from the conductor and ending at infinity, we must get the same result, because the conductor and infinity are both equipotentials. But E drops more quickly from a sharp point or edge than from a smooth surface – as we have seen, E drops as 1/r2 from a point charge, as 1/r near a charged line, and scarcely drops near a charged surface. The integral far from the conductor is similar for all electric field lines; near the conductor, if E·dr drops more quickly from a sharp point or edge, it must be that E starts out larger there. Then, since E is proportional to the surface charge σ, it must be that σ, too, is larger at a sharp point or edge of a conductor.

capacitor battery switch Capacitors and capacitance A capacitor is any pair of isolated conductors. We call the capacitor charged when one conductor has total charge q and the other has total charge –q. (But the capacitor is then actually neutral.) Whatever the two conductors look like, the symbol for a capacitor is two parallel lines.

capacitor battery switch Capacitors and capacitance Here is a circuit diagram with a battery to charge a capacitor, and a switch to open and close the circuit.

capacitor battery switch Capacitors and capacitance The charge q on the capacitor (i.e. on one of the two conductors) is directly proportional to the potential difference ΔV across the battery terminals: q = CΔV.

Capacitors and capacitance The charge q on the capacitor (i.e. on one of the two conductors) is directly proportional to the potential difference ΔV across the battery terminals: q = CΔV. (Note that q and ΔV both scale the same way as the electric field.)

Capacitors and capacitance The charge q on the capacitor (i.e. on one of the two conductors) is directly proportional to the potential difference ΔV across the battery terminals: q = CΔV. (Note that q and ΔV both scale the same way as the electric field.) The constant C is called the capacitance of the capacitor. The unit of capacitance is the farad F, which equals Coulombs per volt: F = C/V.

Capacitors and capacitance Example 1: Parallel-plate capacitor If we can neglect fringing effects (that is, if we can take the area A of the conducting plates to be much larger than the distance d between the plates) then E = σ/ε0 where σ = q/A and ΔV = Ed = qd/ε0A. By definition, q = CΔV, hence C = ε0A/d is the capacitance of an ideal parallel-plate capacitor.

Capacitors and capacitance Example 2: Cylindrical capacitor Again, if we can neglect fringing effects (that is, if we can take the length L of the capacitor to be much larger than the inside radius b of the outer tube) then a b

Capacitors and capacitance Example 3: Spherical capacitor The diagram is unchanged, only E(r) is different: a b

capacitor battery switch Capacitors in series and in parallel Combinations of capacitors have well-defined capacitances. Here are two capacitors in series:

Capacitors in series and in parallel Combinations of capacitors have well-defined capacitances. Here are two capacitors in series: Since the potential across both capacitors is ΔV, we must have q1 = C1ΔV1and q2 = C2ΔV2 where ΔV1+ ΔV2 = ΔV. But the charge on one capacitor comes from the other, hence q1 = q2 = q.

Capacitors in series and in parallel Combinations of capacitors have well-defined capacitances. Here are two capacitors in series: Since the potential across both capacitors is ΔV, we must have q = C1ΔV1and q = C2ΔV2 where ΔV1+ ΔV2 = ΔV. If the effective capacitance is Ceff, then we have hence

Capacitors in series and in parallel For n capacitors in series, the generalized rule is C2 C1 Cn

capacitor battery switch Capacitors in series and in parallel Combinations of capacitors have well-defined capacitances. Here are two capacitors in parallel:

Capacitors in series and in parallel Combinations of capacitors have well-defined capacitances. Here are two capacitors in parallel: Since the potential across each capacitor is still ΔV, the charge on the capacitors is q1 = C1ΔV and q2 = C2ΔV. If the effective capacitance is Ceff, then we have C1ΔV + C2ΔV = q1 + q2 = CeffΔV, thus capacitances in parallel add: C1+ C2 =Ceff .

Capacitors in series and in parallel For n capacitors in parallel, the generalized rule is Ceff = C1+ C2 +…+ Cn . C1 C2 Cn

Halliday, Resnick and Krane, 5th Edition, Chap. 30, MC 9: The capacitors have identical capacitance C. What is the equivalent capacitance Ceff of each of these combinations? A B D C

Halliday, Resnick and Krane, 5th Edition, Chap. 30, MC 9: The capacitors have identical capacitance C. What is the equivalent capacitance Ceff of each of these combinations? Ceff = 2C/3 Ceff = 3C A B Ceff = C/3 Ceff = 3C/2 D C

Halliday, Resnick and Krane, 5th Edition, Chap. 30, Prob. 9: Find the charge on each capacitor (a) with the switch open and (b) with the switch closed. C1 = 1 μF C2 = 2 μF ΔV =12 V C3 = 3 μF C4 = 4 μF

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