Chapter 25 Chapter 25 Electric Potential Electrical Potential Energy! When a test charge, q 0 is placed in an electric field E, it experiences a force: F = q E 0! Is this force conservative? 1 2 Electric Potential Energy! What is the work done by the electric field? Electric Potential Energy, final! Because q o E is conservative, the line integral does not depend on the path taken by the charge! This is the change in potential energy of the system. 3 4 1
Electric Potential! The potential energy per unit charge, U/q o, is the electric potential! The potential is independent of the value of q o! The potential has a value at every point in an electric field U! The electric potential is V = q o Electric Potential, cont.! The potential is a scalar quantity! Since energy is a scalar! As a charged particle moves in an electric field, it will experience a change in potential U B V = = d q E s A o 5 6 Work and Electric Potential! Assume a charge moves in an electric field without any change in its kinetic energy! The work performed on the charge is W = U = q V Units! 1 V = 1 J/C! V is a volt! It takes one joule of work to move a 1- coulomb charge through a potential difference of 1 volt! In addition, 1 N/C = 1 V/m! This indicates we can interpret the electric field as a measure of the rate of change with position of the electric potential 7 8 2
Electron-Volts Illustration by Debra Solomon! Another unit of energy that is commonly used in atomic and nuclear physics is the electronvolt! One electron-volt is defined as the energy a charge-field system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt! 1 ev = 1.60 x 10-19 J 9 10 Potential Difference in a Uniform Field! The equations for electric potential can be simplified if the electric field is uniform: Problem 1 2. A uniform electric field of magnitude 325 V/m is directed in the negative y direction as shown in the figure. The coordinates of point A are (-0.200, -0.300) m, and those of point B are (0.400, 0.500) m. Calculate the potential difference V B V A, using the blue path. 11 12 3
Energy and the Direction of Electric Field PE is + PE is Problem 2 1. Calculate the speed in km/s of a proton that is accelerated from rest through a potential difference of 133 V. 13 14 More About Directions! A system consisting of a positive charge and an electric field loses electric potential energy when the charge moves in the direction of the field! An electric field does work on a positive charge when the charge moves in the direction of the electric field! The charged particle gains kinetic energy equal to the potential energy lost by the charge-field system! Another example of Conservation of Energy 15 Directions, cont.! If q o is negative, then U is positive! A system consisting of a negative charge and an electric field gains potential energy when the charge moves in the direction of the field! In order for a negative charge to move in the direction of the field, an external agent must do positive work on the charge 16 4
Equipotentials! Point B is at potential than point A! Points B and C?! The name equipotential surface is given to any surface consisting of a continuous distribution of points having the same electric potential Potential and Point Charges! A positive point charge produces a field directed radially outward! The potential difference between points A and B is 17 18 Potential and Point Charges, cont.! The electric potential is independent of the path between points A and B! It is customary to choose a reference potential of V = 0 at r A =! Then the potential at some point r is V = q ke r Potential Energy of Multiple Charges! Consider two charged particles! The potential energy of the system is U = k e qq r 1 2 12 19 20 5
More About U of Multiple Charges U with Multiple Charges, final! If the two charges are the same sign, U is positive and work must be done to bring the charges together! If the two charges have opposite signs, U is negative and work is done to keep the charges apart 21! If there are more than two charges, then find U for each pair of charges and add them! For three charges: qq 1 2 qq 1 3 qq 2 3 U = ke + + r12 r13 r23! The result is independent of the order of the charges 22 E and V for an Infinite Sheet of Charge are the dashed blue lines! The electric field lines are the brown lines are everywhere perpendicular to the field lines E and V for a Point Charge are the dashed blue lines! The electric field lines are the brown lines are everywhere perpendicular to the field lines 23 24 6
E and V for a Dipole are the dashed blue lines! The electric field lines are the brown lines are everywhere perpendicular to the field lines Electric Field from Potential, General! In general, the electric potential is a function of all three dimensions! Given V (x, y, z) you can find E x, E y and E z as partial derivatives V V V Ex = Ey = Ez = x y z 25 26 Problem 3 3. Over a certain region of space, the electric potential is V = 2x - x 2 y + 2yz 2. Find the expression for the x, y, and z component of the electric field over this region. (Use x, y, and z as necessary.) V for a Continuous Charge Distribution (Extra material)! To find the total potential, you need to integrate to include the contributions from all the elements dq V = ke r! This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions 27 28 7
V for a Uniformly Charged Ring (Extra material)! P is located on the perpendicular central axis of the uniformly charged ring! The ring has a radius a and a total charge Q dq kq e V = ke = r x + a 2 2 V for a Uniformly Charged Disk (Extra material)! The ring has a radius a and surface charge density of σ ( ) 1 = + 2 2 2 V 2πkeσ x a x 29 30 V for a Finite Line of Charge (Extra material)! A rod of line l has a total charge of Q and a linear charge density of λ 2 2 kq e! +! + a V = ln! a V for a Uniformly Charged Sphere (Extra material)! A solid sphere of radius R and total charge Q Q! For r > R, V = ke r! For r < R, kq e 2 2 VD VC = R r 3 2R ( ) 31 32 8
Electrostatic Precipitator The Xerographic Process! 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 33 34 Application Laser Printer Potentials Due to Various Charge Distributions! The steps for producing a document on a laser printer is similar to the steps in the xerographic process! Steps a, c, and d are the same! The major difference is the way the image forms on the selenium-coated drum! A rotating mirror inside the printer causes the beam of the laser to sweep across the selenium-coated drum! The electrical signals form the desired letter in positive charges on the selenium-coated drum! Toner is applied and the process continues as in the xerographic process 35 36 9