Electric and Magnetic Potential Flashcards
Electric Potential Energy
Definition
-electric potential energy is defined as:
dU = - |F . d|L = -q|E . d|L
Potential Difference
Definition
-the definition of electric potential gives the potential difference between two points as:
ΔV = b,a ∫ -|E . d|L
-the potential difference between two points is often termed the voltage
Which forces can potential energies be defined for?
- potential energies can only be defined for conservative forces
- for a conservative force the work done is independent of the path taken, this means that certain points in space can be associated with a particular potential energy
- this is not the case for non-conservative forces for which the work done does depend on the path taken
Electric Potential a Distance R From a Single Isolated Point Charge
Derivation
-place the charge q at the origin of the coordinate system and dV is given by:
dV = -|E . d|L
-for a point charge, |E=kq/r² ^r
-since V is independent of path taken, choose the easiest path i.e. moving radially
d|L = dr ^r
-this gives
dV = -kq/r² ^r . dr ^r
dV = -kq/r² dr
-integrate
ΔV = b,a ∫-kq/r² dr
ΔV = [kq/r] b,a
-consider moving from an infinite distance to a distance R
ΔV = [kq/r] R,∞ = kq/R - 0 = k q/R
-here we have calculated the change in potential, not the potential itself
Electric Potential Due to a Set of Discrete Charges
Derivation
-for a single point charge we have
V = kq/r (taking V=0 at infinity)
-for a set of N discrete charges, the superposition principal gives:
|E(|r) = i=1->N Σ |E(qi,|ri) and |E(qi, |ri) = k qi/ri² ^ri
-where |ri is the vector from the point charge i to the point in space r where we are measuring |E
dV = -|E . d|L
dV = i=1->N -Σ [k qi/ri² ^ri] . dr ^r
-integrate
ΔV = b,a ∫ dV = b,a -∫ i=1->N Σ [k qi/ri² ^ri] . dr ^r
-take the electric potential at infinity to be 0
V = - b,x∫ [k q1/r1² ^r1] . dr ^r - b,x∫ [k q2/r2² ^r2] . dr ^r -…- b,x∫ [k qN/rN² ^rN] . dr ^r
V = V1 + V2 + V3 + … + VN
V = k q1/R1 + k q2/R2 + … + k qN/RN
-the superposition of electric potentials
Electric Potential at Infinity
- from the derivation of the change in electric potential on moving from an infinite distance to a distance R from a point charge it may appear that we have shown that the electric potential at infinity is zero
- we have actually calculated the change in electric potential
- you can define the electric potential at infinity to be any fixed value without affecting the physics
- usually, zero is mathematically the easiest choice
Superposition of Electric Potentials
-for a set of N discrete charges
V = k q1/R1 + k q2/R2 + k q3/R3 + … + k qN/RN
V = k i=1->N Σ qi/Ri
Electric Potential for a Continuous Charge Distribution
Derivation
-for a discrete charge distribution: V = k i=1->N Σ qi/Ri -the summation becomes an integral over each tiny element dq of charge V = k ∫dq/R -for a continuous charge distribution, dq is given by dq = ρdV -for V this gives: V = k ∫ρdV/R
Electric Potential and Electric FIeld
-we defined V in terms of electric field as:
dV = -|E . d|L
dV = -(Ex^i + Ey^j + Ez^k).(dx^i +dy^j + dz^k)
dV = -Exdx - Eydy - Ezdz
-this gives electric field components in terms of electric potential as:
Ex = -∂V/∂x , Ey = -∂V/∂y , Ez = -∂V/∂z
-so electric field as a vector in terms of electric potential is:
|E = -(∂V/∂z ^i + ∂V/∂y ^j + ∂V/∂z ^k)
|E = -|∇ V
-i.e. E is -grad V
Meaning of Grad
-grad points in the direction of greatest increase of the field
so |E = - |∇ V means that |E point in the direction of greatest decrease of electric potential
Circulating Electric Fields and Electric Potential Energy
|E = - |∇ V and from Faraday’s Law, |∇x|E = -∂|B/∂t
- this means that we only have circulating electric fields due to fluctuating magnetic fields
- these induced electric fields are not conservative so it makes no sense to describe them using a potential energy function
Curl of an Electrostatic Field
-for an electric field due purely to a set of stationary charges, an electrostatic field, Faraday’s Law gives:
|∇x|E = 0
Curl of the Gradient
-the curl of the gradient is always zero:
-consider I = |∇x|∇ψ
I = |∇ x (∂ψ/∂x ^i + ∂ψ/∂y ^j + ∂ψ/∂z ^k)
I = (∂²ψ/∂y∂z - ∂²ψ/∂z∂y) ^i + (∂²ψ/∂z∂x - ∂²ψ/∂x∂z) ^j + (∂²ψ/∂x∂y - ∂²ψ/∂y∂x ) ^k
I = 0 ^i + 0 ^j + 0 ^k
I = 0
-this makes sense as you cant start at a point in a field and move in a loop always in the direction of increasing field and end up back where you started
Poisson’s Equation
Derivation
-relationship between electrostatic field and electric potential:
|E = - |∇ V
-the differential form of Gauss’s Law states that:
|∇ . |E = ρ/εo
-combining these equations gives:
|∇ . (- |∇ V) = ρ/εo
|∇ . |∇V = ∇²V = - ρ/εo
Poisson’s Equation
Equation
|∇ . |∇V = ∇²V = - ρ/εo
