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Physics for Engineers II Lecture 3 Electric potential energy Point charge potential capacity for doing work (arises from position or configuration) any collection of charges The change in potential is the work done per unit charge against the Coulomb force, so For a positive charge q the Coulomb force does positive work in moving it from a to b, so that represents a drop in potential energy. more convenient electric potential or voltage Potential reference at infinity Taking the limit as rb → ∞ gives for any arbitrary value of r. The choice of potential equal to zero at infinity is logical in this case because the electric field and force approach zero there. the work necessary to bring a test charge q from an infinite distance to some distance r M.Mulak / WUT Zero potential physical significance change in potential zero point is arbitrary like the origin of a coordinate system set for convenience some physical or geometric logic to the choice of the zero point A single point charge or localized collection of charges the zero point at infinity Practical electrical circuits the earth or ground potential is taken to be zero 1 Physics for Engineers II Lecture 3 Equipotential lines Multiple point charges any number of point charges contour V calculated from the point charge expression by simple addition (a scalar quantity!) in this case the "altitude" is electric potential or voltage lines on a map which trace lines of equal altitude - always perpendicular to the electric field - 3D equipotential surfaces - movement along an equipotential surface requires no work simpler than the vector sum required to calculate the electric field! Electric dipole potential Dipole moment superposing the point charge potentials of the two charges a pair of opposite charges of magnitude q Useful concept: - in atoms and molecules the effects of charge separation are measurable, but the distances between the charges are too small to be easily measurable - in dielectrics and other applications in solid and liquid materials M.Mulak / WUT 2 Physics for Engineers II Lecture 3 Potential of line charge a continuous charge distribution can be found by superposing the point charge potentials of infinitesimal charge elements Potential for ring of charge a continuous charge distribution each element of the ring has the same distance r from the point P The ring potential can then be used as a charge element to calculate the potential of a charged disc. If the charge is characterized by an area density and the ring by an incremental width dR', then: Potential for disc of charge by summing the potentials of charged rings (the integral over the charged disc) M.Mulak / WUT Electric field from voltage Great value of the scalar electric potential V E The component of electric field in any direction is the negative of rate of change of the potential in that direction. 3 Physics for Engineers II Lecture 3 Electric field as gradient Voltage difference and electric field the change in voltage = the work done per unit charge against the electric field. a constant electric field more general case M.Mulak / WUT 4 Physics for Engineers II Lecture 3 Potential of a charged conducting sphere Gauss' law the electric field outside the sphere is identical to that of a point charge, so Applications of electrostatics SCIENTIFIC Electrostatic generators (Van de Graaff generator): used for acceleration of elementary particles (e.g. in nuclear physics) PRACTICAL E=0 inside, so V=const at the value it reaches at the surface Lightning arrester Electrostatic precipitators: used to reduce atmospheric pollution Xerography The idea of a lightning arrester Model of a sharp point conductor at the same V r1 Near the spike r 0 a strong E ionises the air and make electrical breakdown more likely to occur q1 E= M.Mulak / WUT σ ε0 q2 conducting wire V = ke E1 = k e From Gauss’s Law r2 q1 q = ke 2 r1 r2 q1 2 r1 r2 → 0 ⇒ ⇒ E2 = k e E1 →0 E2 q1 r1 = q2 r2 q2 E r ⇒ 1 = 2 2 E2 r1 r2 when E1 is finite ⇒ E 2 → ∞ 5 Physics for Engineers II Lecture 3 Van de Graaff Generator Voltages of hundreds of thousands of volts Discharges do not represent a serious shock hazard since the currents attainable are small Voltages enough to make our hair stand on end. Like charges on individual hairs makes them repel each other and stand away from Jennifer's head. She is standing on a wooden chair to isolate her from ground so that the charge will build up on her. Electrostatic precipitator: an application of electrical discharge in The xerographic process gases Idea: a photoconductive material Used to remove dust from combustion gases (efficiency up to 99%)) THE BASIC IDEA: High voltage (40 kV ÷100 kV) between the wire running down the center of the duct and the outer wall (grounded). The wire is at negative potential with respect to the walls ⇒ E toward the wire Corona discharge around the wire (the highest E) ⇒ ionization of the air ⇒ negative ions accelerated toward the outer wall Due to collisions the dirt particles in the streaming gas become charged (mostly negatively) ⇒ they are attracted to the outer wall Periodically shaking the duct ⇒ the ash and dust particles fall down and are collected. M.Mulak / WUT (a) The photoconductive surface is positively charged. (b) Using light an image is formed on the surface in the form of hidden positive charges. (c) The surface containing the image is covered with a negatively charged powder (toner), which adheres only to the image area. (d) A piece of paper is placed over the surface and given a charge. This transfers the (e) A laser printer idea visible image to the paper, which is then heat-treated to „fix” the powder to the paper. 6