Chapter 22 Electric Potential (Voltage)

Chapter 22 Electric Potential (Voltage)

Question 29.5 Work and Electric Potential I Which requires the most work, to move a positive charge from P to points 1, 2, 3 or 4? All points are the same distance from P. 1) P 1 2) P 2 3) P 3 4) P 4 5) all require the same amount of work 3 2 1 P E 4

Electric potential energy Recall how a conservative force is related to the potential energy associated with that force: The electric potential energy is the potential energy due to the electric force, which can be expressed in terms of the electric field. If location A is chosen to be the zero point, then the electric potential energy at location B (which we now call r) is given by Potential energy of particle is a scalar function of space.

Consider uniform electric field (say inside a parallel capacitor) If a proton is taken from location B to location C, how does its potential energy change? 1. it decreases 2. it increases 3. it doesn t change

Suppose a proton is released from rest just below the top (positive) plate of an parallel plate capacitor with an electric field strength E = 100 N/C. If the distance between the plates is d = 3 mm, how fast is it moving when it hits the bottom (negative) plate?

Electric Potential energy of two Point Charges What is the change in potential energy of the test charge as it goes from position a to position b? U AB = B A q 0 E dr U AB = rb r a q 0 kq r 2 dr U AB = kq 0 q r r b r a = k q 0q r b k q 0q r a If we let r a be infinity (the zero point), and r b an arbitrary distance, then U(r) =k q 0q r

Example Rutherford scattering. A helium nucleus of mass 4 m p is emitted with an initial speed of v 0 = 4.9 x 10 5 m/s towards a gold nucleus of charge q 2 = 79 e. What is the minimum distance between the two particles (assume the gold nucleus doesn t move)?

Electric Potential (Voltage) Electric potential, or voltage, at a point in space is defined as the electric potential energy per unit charge associated with a test charge at that point. V (r )= U(r ) q Potential energy deals with the energy of a particle. Voltage deals with all locations in space (no particle needs to be there). Analogous to how a particle experiences a force, but an electric field can exist at any point in space. Unit of electric potential is the volt (V). 1 V = 1 J/C.

Potential Difference (Voltage Difference) Voltage difference is defined as Because the electrostatic field is conservative, it doesn t matter what path is taken between those points. In a uniform field, the potential difference becomes V AB = E r

Clicker Question In a parallel plate capacitor, the electric field is uniform and is directed from the positive plate to the negative plate. An electron goes from location A to location C. Which statement is true? A) The electron goes from a high voltage to a lower voltage. B) The electron goes from a low voltage to a higher voltage. C) The voltage is the same at both locations.

Clicker question The figure shows three straight paths AB of the same length, each in a different electric field. Which one of the three has the largest magnitude of a voltage difference between the two points? A. (a) B. (b) C. (c)

Millikan s Oil Drop Experiment Charged oil droplets made to levitate inside capacitor Measure voltage difference across plates Release and measure terminal velocity (which gives droplet radius/mass) Determine net charge on droplet.

Voltage of a point charge Recall the potential energy of two point charges: U(r) =k q 0q r Thus the voltage a distance r from the charge q is given by V (r) =k q r (there is no test charge anymore)

Voltage due to a charge distribution If the electric field of the charge distribution is known, the voltage can be found by integration. Alternatively, the voltage can be found by summing point-charge potentials: For discrete point charges, V (P )= V ( P)= i kq i r i For a continuous charge distribution, ( ). V P E dr = k dq = r i E i dr = i E i dr = i V i (P )

Clicker Question Two identical positive charges of charge Q are a distance d apart. What is the voltage at the midway point between the charges? a) k Q/d b) 2 k Q/d c) 4 k Q/d d) 8 k Q/d e) 0

Clicker question Location P is equidistant from the two charges of an electric dipole. The voltage at P is a) positive b) zero c) negative

Question 29.10 Hollywood Square Four point charges are arranged at the corners of a square. Find the electric field E and the potential V at the center of the square. 1) E = 0 V = 0 2) E = 0 V 0 3) E 0 V 0 4) E 0 V = 0 5) E = V regardless of the value -Q +Q -Q +Q

Clicker Question A solid sphere of radius R has a UNIFORM charge density per unit volume ρ and net charge Q. The voltage at the center of the sphere is 1. V = k Q/R 2. V < kq/r 3. V > kq/r

Maximum voltage of a Van de Graaff generator. Molecules in air get ionized for electric fields greater than roughly E max = 3 x 10 6 V/m. What is the maximum voltage of a charged sphere of radius R=0.2 m?

Voltage due to a charged ring For a uniformly charged ring of total charge Q, integration gives the potential on the ring axis: kdq V = dq = λadθ r 2π kλa dθ V (x, y, z) = r(θ,x,y,z) Very hard integral in general! If P is on x axis, then r only depends on x and a. 0 On x axis: kdq k kq r r x + a ( ) = = dq= 2 2 V x

Voltage due to a long charged wire Find the voltage a distance r from a very long line of charge with linear charge density λ and radius R since this is an infinitely extended object, we can t use infinity as a zero point. Instead, let s say V(r=R) = 0.

Equipotentials An equipotential is a surface on which the potential (voltage) is constant. In two-dimensional drawings, we represent equipotentials by curves similar to the contours of height on a map. The electric field is always perpendicular to the equipotentials. ( V = E s = 0)

Clicker question The figure shows cross sections through two equipotential surfaces. In both diagrams the potential difference between adjacent equipotentials is the same. Which of these two could represent the field of a point charge? A. (a) B. (b) C. neither (a) nor (b)

Conductors There s no electric field inside a conductor in electrostatic equilibrium. At the surface there s no parallel component of the electric field. Therefore in electrostatic equilibrium, the entire conductor is at the same potential

Determining E from V Voltage can be determined if electric field is known Can electric field be determined if voltage is known? For a very small displacement, V = E dr = E r Suppose Then E r = E x x E x = V x = V x Can do the same thing in other direction: E = V = r = x î V x î + V y ĵ + V z ˆk The derivatives here are partial derivatives, expressing the variation with respect to one variable alone. (gradient of V)

For which region is the magnitude of the electric field the highest? 9 A 200 V B 180 V 160 V 140 V 120 V 1. A 2. B 3. C 4. D 8 Distance (cm) 7 6 5 4 C D 100 V 80 V 3 2 1 1 2 3 4 5 6 7 8 9 10 Distance (cm)

CT 29.13b What is the approximate magnitude of the electric field at point A? (Each equipotential line is 2 m from the nearestneighbor equipotential.) A) 0.1 Volts/m B) 0.2 Volts/m C) 1.6 Volts/m D) 0.7 Volts/m E) None of these A 0V -1.4V -1.8V -2.1V

Example: Electric field along axis of charged Ring: Recall that the voltage due to a charged ring is: kdq k kq r r x + a ( ) = = dq= 2 2 V x Use this to determine E(x):