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3.1.4.     Electric Potential & Potential Energy <CJ chap 19.1-19.4  >

3.1.4.1. Discussion

3.1.4.1.1.     The potential energy of a system of charges is the work necessary to assemble them from infinity

3.1.4.1.1.1. The potential energy, U, is a scalar and is measured in units of Joules

3.1.4.1.2.     The electric potential V(r), is the work needed to bring a unit charge to this point from infinity

3.1.4.1.2.1. V(r) is also a scalar and is measured in units of Volts = Joule / Coulomb

3.1.4.1.2.2. The plotting of  the equal potential lines V(r) = constant for a system displays contours of V

3.1.4.1.2.3. These contours are always exactly perpendicular to the electric field E lines everywhere

3.1.4.1.2.4. In fact E is equal to (the negative of ) the gradient (rate and direction of maximum change) of V

3.1.4.1.2.5. Constant V(r) curves are good visual representations of the electrostatic environment, as is E

3.1.4.1.2.6. It is most common to consider changes in V (voltage differences) rather than absolute values

3.1.4.2. Mathematical:

3.1.4.2.1.     Potential Energy =  U = k q₁ q₂ / |r₁ - r₂|  = Work needed to bring q₁ & q₂ from an infinite distance

3.1.4.2.1.1. The units of potential energy here are Joules.  Note that U is a scalar not a vector.

3.1.4.2.1.2. The potential energy of several charges, q_i is given by U = ½  k Sq_i q_j / |r_i-r_j| 

3.1.4.2.1.2.1.     note the ½ arises from double counting in the summation over i and j

3.1.4.2.2.     Electric Potential = V(r) = U/q₀  = the work needed to bring a unit charge q₀ from infinity to the point r

3.1.4.2.2.1. Thus  V(r) = k q / r  at r due to a charge q at the origin   

3.1.4.2.2.2. The units of electric potential are given in Volts = Joules / Coulomb  (or V=J/Q)

3.1.4.2.2.3. Usually, we look at voltage differences such as the potential difference between battery terminals.

3.1.4.2.3.     Equipotential lines (curves that follow equal potential values) are perpendicular everywhere to E

3.1.4.2.3.1. These equipotential curves can be compared to isotherms (temperature) or isobars (pressure).

3.1.4.3. Advanced:

3.1.4.3.1.     Potential Energy = Work = dU = F dr = -q₁ E dr    =  -k q₁ q₂ dr₁₂ / r₁₂²  

3.1.4.3.1.1. Thus U = k q₁ q₂ / r₁₂    where r₁₂ = |r₁-r₂|  and when the integral goes from infinity up to  r₁₂ 

3.1.4.3.1.2. The units of potential energy U are in Joules and U is a scalar as it is a dot product

3.1.4.3.2.     Electric Potential = V = U/q   or for a single charge at the origin,   V(r) = k q / r   

3.1.4.3.2.1. The units of V are in Volts (V) where V=J/Q

3.1.4.3.2.2. Since DV = E dr  then it follows that E_x =  and generally that E = V

3.1.4.3.2.3. One recalls that