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2/9/17 Electric Potential (3) A. B. Kaye, Ph.D. Associate Professor of Physics 14 February 2017 Update • Last class • • Continued Electric Potential Homework #2 was due • Today • • Complete Electric Potential Homework #3 Due • Tomorrow • Exam I ELECTRIC POTENTIAL Potential Difference due to a Charged Conductor 1 2/9/17 V Due to a Charged Conductor • Consider two points on the surface of the charged conductor in the sketch • E is always perpendicular to the displacement ds • Therefore, , and the potential difference between A and B is also zero V Due to a Charged Conductor, cont. • V is constant everywhere on the surface of a charged conductor in equilibrium • DV = 0 between any two points on the surface • The surface of any charged conductor in electrostatic equilibrium is an equipotential surface • I.e., Every point on the surface of a charged conductor in equilibrium is at the same electric potential • Because the electric field is zero inside the conductor, we conclude that: • • The electric potential is constant everywhere inside the conductor The electric potential is equal to the value at the surface Irregularly Shaped Objects • For irregularly-shaped objects, the charge density is: • • high where the radius of curvature is small low where the radius of curvature is large • The electric field is: • • large near the convex points having small radii of curvature reaches very high values at sharp points 2 2/9/17 Cavity in a Conductor • Create an irregularlyshaped cavity inside a conductor • Assume no charges are inside the cavity • The electric field inside the conductor must be zero Cavity in a Conductor, cont • The electric field inside does not depend on the charge distribution on the outside surface of the conductor • For all paths between A and B, • A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity Corona Discharge • If the electric field near a conductor is sufficiently strong, electrons resulting from random ionizations of air molecules near the conductor accelerate away from their parent molecules • These electrons can ionize additional molecules near the conductor • This creates more free electrons • The corona discharge is the glow that results from the recombination of these free electrons with the ionized air molecules • The ionization and corona discharge are most likely to occur near very sharp points 3 2/9/17 ELECTRIC POTENTIAL The Millikan Oil Drop Experiment Millikan Oil Drop Experiment • In 1909, Robert Millikan and Harvey Fletcher measured e, the magnitude of the elementary charge on the electron • They also demonstrated the quantized nature of this charge • Their experiment entailed observing tiny charged droplets of oil between two horizontal metal electrodes and the droplets are illuminated by a light to be observed • Let’s look at the experiment in more detail Millikan Oil Drop Experiment – Experimental Set-Up 4 2/9/17 Oil Drop Experiment • First, without any applied electric field, the terminal velocity of a droplet was measured • • • At the terminal velocity, the drag force is equal to the gravitational force These depend on the radius in different ways, so that the radius of the droplet, and therefore the mass and gravitational force, could be determined From Stokes’ Law and Newton’s 2nd Law, the drop reaches a terminal velocity and we can compute its size 5 2/9/17 Oil Drop Experiment – Drop Radius • From Stokes’ Law, we know: • For a spherical droplet, we know that the gravitational force is: • When the drop is not accelerating, the forces balance (FD = FG), implying Oil Drop Experiment • In the second part of the experiment, an adjustable voltage was applied between the plates to induce an electric field • An X-ray source was turned on to induce a charge on each oil drop • The voltage was adjusted until the drops were suspended in mechanical equilibrium New Force • With the electric field, the force on the drop is now: • For parallel plates with a potential difference V and a separation d, the electric field is: • To solve for q, simply adjust V until the drop is stationary, at which point, FE and w are equal: 6 2/9/17 Final Solution(s)… • This would mean that • If, instead, you adjusted V so that the drop has a new terminal velocity, then Oil Drop Experiment • The drop can be raised and allowed to fall numerous times by turning the electric field on and off • After many experiments, Millikan and Fletcher determined that • • • This experiment yielded conclusive evidence that charge is quantized ELECTRIC POTENTIAL Examples of Applications of Electrostatics 7 2/9/17 Van de Graaff Generator • Charge is delivered continuously to a high-potential electrode by means of a moving belt of insulating material • The high-voltage electrode is a hollow metal dome mounted on an insulated column • Large potentials can be developed by repeated trips of the belt • Protons accelerated through such large potentials receive enough energy to initiate nuclear reactions Electrostatic Precipitator • An application of electrical discharge in gases is the electrostatic precipitator • It removes particulate matter from combustible gases • The air to be cleaned enters the duct and moves near the wire • As the electrons and negative ions created by the discharge are accelerated toward the outer wall by the electric field, the dirt particles become charged • Most of the dirt particles are negatively charged and are drawn to the walls by the electric field CAPACITANCE AND DIELECTRICS “Interesting Problems to Solve” 8 2/9/17 Interesting Problem #11 • Suppose that near the ground directly below a thundercloud, the electric field is of a constant magnitude 2.0 x 104 V/m and points upward. What is the potential difference between the ground and a point in the air 50 m above the ground? Interesting Problem #12 Assume that the electron in a hydrogen atom is 5.3 x 10–11 m from the proton, and assume that the proton is a small ball of charge with q′ = 1.60 x 10–19 C. Find the electrostatic potential generated by the proton at this distance and then determine the potential energy of the electron. Interesting Problem #13 A sphere of radius R carries a total positive charge Q distributed uniformly throughout its volume. Find the electrostatic potential both inside and outside the sphere. 9 2/9/17 Interesting Problem #14 A coaxial cable consists of a long, cylindrical conductor of radius a concentric with a thin cylindrical shell of larger radius b. If the central conductor has a charge per unit length l = Q/L uniformly distributed on its surface, what is the potential difference between the inner and outer conductors? (Assume the space between them is empty.) Interesting Problem #15 Four charges, q1 = q, q2 = 2q, q3 = –q, and q4 = q, are at the corners of a square with side length a. If q = 2.0 µC and a = 7.5 cm, what is the total energy required to assemble this system of charges? Interesting Problem #16 The metallic sphere on the top of a large Van de Graaff generator has a radius of 3.0 m. Suppose that the sphere carries a charge of 5.0 x 10–5 C uniformly distributed over its surface. How much electric energy is stored in this charge distribution? 10 2/9/17 CAPACITANCE AND DIELECTRICS Key Review Points Key Review Points • The potential difference between points A and B in an electric field E is defined as • The electric potential V = U/q is a scalar quantity with units of volts where • An equipotential surface is a surface on which all points have the same electric potential • • Equipotential surfaces are perpendicular to electric field lines The potential difference between two points separated by a distance d in a uniform electric field E is Key Review Points • When a positive charge q is moved between points A and B in an electric field E, the change in the potential energy of the charge-field system is • If we define V = 0 at r = ∞, the electric potential due to a point charge at any distance r from the charge is • The electric potential energy associated with a pair of point charges separated by a distance r12 is • For a system of point charges, the total potential energy is the sum of the individual charges 11 2/9/17 Key Review Points • The electric potential due to a continuous charge distribution is • The individual components of the electric field can be found by: • Potential energy of a conductor: • • For a system of conductors, the total energy is the sum of the components Energy density in an electric field: 12