Electric Potential Energy • The electric field is like the gravitational field: it is a conservative field• . This means that we can associate with it an electric • potential energy U. Further, (kinetic energy K + electric potential energyU) are then conserved if electric forces are the only forces operating on the studied system • If the electric field exerts a force during a move from an initial point F on a particle b, the change in potential energy is a to a final point Ub−Ua=ΔU=−Wa→bPHYS 1003 Electricity and Magnetism L6 23-1 1 Electric Potential Energy (2) • The work done by the force during the displacementfrom a to b is bb Wa→b=∫F⋅dl=∫Fcosφdlaa• Here φ denotes the angle between the electric force F and the displacement vector dl. PHYS 1003 Electricity and Magnetism L6 23-1 2 PHYS 1003 Electricity and Magnetism - L6 1 Electric Potential Energy (3) For a • uniform electric field E: When a charge moves a displacement same direction as the field d in the So Wa→b=Fd=qoEd Ub−Ua=−(qoEd)Hence, if the charge q 0 is positive, Ub
Ua has to be done to ‘lift’ the charge against the field PHYS 1003 Electricity and Magnetism L6 23-1 4 PHYS 1003 Electricity and Magnetism - L6 2 Electric Potential Energy of a Point Charge (non-uniform E) Radial component of the electrostatic force is Fqqor=Hence, the work done by the 4πεor2electrostatic force going from a to b is rbrbWqqoa→b=or ∫Frdr=ra ∫ra4πε2drorWqqo& 11) a→b=4πε( −+ o' rarb* These are consistent with the electrostatic energy at radius r being U=qqo 4πεorPHYS 1003 Electricity and Magnetism L6 23-1 5 Electric Potential Energy of Several Point Charges The potential energy associated with a charge q0 due to a collection of other charges q1, q2, q3 ,……,is the algebraic (not vector!) sum of the pairwise potentials: U=qoqi4πεoi=∑1,2,...riThis is the potential energy due to the interaction of q0 with all the other charges. Taking into account all interactions among charges: U=1qiqj4πεo∑i