111/28/2015 ELECTRICITY AND MAGNETISM Phy 220 Chapter 3: Electric Potential

211/28/ Potential energy of a uniform electric field The electrostatic force is conservative ( W done by it independent of path ) As in mechanics, work is Work done on the positive charge by moving it from A to B: Work done on the positive charge by moving it from A to C: AB d C L ϑ

311/28/ Potential energy of electrostatic field The work done by a conservative force equals the negative of the change in potential energy,  U This equation is valid only for the case of a uniform electric field is valid only for the case of a uniform electric field allows to introduce the concept of electric potential allows to introduce the concept of electric potential

411/28/ Electric potential The potential difference between points A and B, V B -V A, is defined as the change in potential energy (final minus initial value) of a charge, q, moved from A to B, divided by q Electric potential is a scalar quantity Electric potential is a measure of electric energy per unit charge (V=U/q) Potential is often referred to as “voltage” Note that, in the previous figure,

511/28/2015 Electric potential - units Electric potential difference between 2 points A and B is the work done to move a charge from B to A divided by the charge. Thus the SI units of electric potential In other words, 1 J of work is required to move a 1 C charge through a potential difference of 1 V If one returns to the case of uniform electric field the potential difference in the direction of the field will be:

611/28/2015 The electric potential decreases in the direction of the electric field.  The electric potential decreases in the direction of the electric field. (a) When the electric field is directed downward, point B is at a lower electric potential than point A. As a positive test charge moves from A to B, the electric potential energy decreases. (b) An object of mass m moves in the direction of the gravitational field, the gravitational potential energy decreases.

711/28/ Electric potential and potential energy due to point charges Electric circuits: point of zero potential is defined by grounding some point in the circuit Electric potential due to a charged particle at a point in space: point of zero potential is taken at an infinite distance from the charge With this choice, the electric potential created by a point charge q at any distance r from q is given by

811/28/2015 Superposition principle for potentials If more than one point charge is present, their electric potential can be found by applying superposition principle The total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges. Remember that potentials are scalar quantities!

911/28/2015 Potential energy of a system of point charges Consider a system of two particles If V 1 is the electric potential due to charge q 1 at a point P, then work required to bring the charge q 2 from infinity to P without acceleration is q 2 V 1. If the distance between P and q 1 is r, then by definition Potential energy is positive if charges are of the same sign and vice versa. PA q1q1 q2q2 r

1011/28/2015 The electron volt A unit of energy commonly used in atomic, nuclear and particle physics is electron volt (eV) The electron volt is defined as the energy that electron (or proton) gains when accelerating through a potential difference of 1 V Relation to SI: 1 eV = 1.60  C·V = 1.60  J V ab =1 V

1111/28/2015 Problem-solving strategy Remember that potential is a scalar quantity Superposition principle is an algebraic sum of potentials due to a system of charges Signs are important Just in mechanics, only changes in electric potential are significant, hence, the point you choose for zero electric potential is arbitrary.

1211/28/2015 Example : ionization energy of the electron in a hydrogen atom  In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5.29  m. Find the ionization energy of the atom, i.e. the energy required to remove the electron from the atom. Note that the Bohr model, the idea of electrons as tiny balls orbiting the nucleus, is not a very good model of the atom. A better picture is one in which the electron is spread out around the nucleus in a cloud of varying density; however, the Bohr model does give the right answer for the ionization energy

1311/28/2015 In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5.29 x m. Find the ionization energy, i.e. the energy required to remove the electron from the atom. Given: r = x m m e = 9.11  kg m p = 1.67  kg |e| = 1.60  C Find: E=? The ionization energy equals to the total energy of the electron-proton system, The velocity of e can be found by analyzing the force on the electron. This force is the Coulomb force; because the electron travels in a circular orbit, the acceleration will be the centripetal acceleration: with or Thus, total energy is

1411/28/ Equipotential surfaces They are defined as a surface in space on which the potential is the same for every point (surfaces of constant voltage) The electric field at every point of an equipotential surface is perpendicular to the surface convenient to represent by drawing equipotential lines