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Electric potential energy • Recall how a conservative force is related to the
potential energy associated with that force:
• The electric potential energy is the potential energy due to the electric force, which can be expressed in terms of the electric field.
• If location A is chosen to be the zero point, then the electric potential at location B (which we now call r) is given by
Potential energy of particle is a scalar function of space.
Consider uniform electric field (say inside a parallel capacitor)
If a proton is taken from location B to location C, how does its potential energy change?
1. it decreases 2. it increases 3. it doesn’t change
Suppose a proton is released from rest just below the top (positive) plate of an parallel plate capacitor with an electric field strength E = 100 N/C. If the distance between the plates is d = 3 mm, how fast is it moving when it hits the bottom (negative) plate?
Electric Potential (Voltage) • Electric potential, or voltage, is defined as the electric
potential energy per unit charge a test particle would have if it were located at a position
• Potential energy deals with the energy of a particle. Voltage deals with all locations in space (no particle needs to be there).
• Analogous to how a particle experiences a force, but an electric field can exist at any point in space.
• Electric Potential difference is defined as
• Because the electrostatic field is conservative, it doesn’t matter what path is taken between those points.
• In a uniform field, the potential difference becomes
Clicker Question In a parallel plate capacitor, the electric field is uniform and is directed from the positive plate to the negative plate. An electron goes from location A to location C. Which statement is true?
A) The electron’s potential energy increases and it goes to a region of higher voltage.
B) The electron’s potential energy decreases and it goes to a region of lower voltage.
C) The electron’s potential energy increases and it goes to a region of lower voltage.
Clicker question • The figure shows three straight paths AB of the same
length, each in a different electric field. Which one of the three has the largest potential difference between the two points?
A. (a) B. (b) C. (c)
Electric potential of a point charge • The point-charge field varies with
position, so we need to integrate:
• Taking the zero of potential at infinity and letting gives
Rutherford scattering. A helium nucleus of mass 4 mp is emitted with an initial speed of v0 = 4.9 x 105 m/s towards a gold nucleus of charge q2 = 79 e. What is the minimum distance between the two particles (assume the gold nucleus doesn’t move)?
Example
Electric potential of a charge distribution • If the electric field of the charge distribution is known,
the electric potential can be found by integration. • The electric potential can always be found by summing
point-charge potentials: • For discrete point charges,
• For a continuous charge distribution,
Potential of charged sphere • Outside sphere, electric field is identical to that of a point
charge. Inside, E=0.
• What is V for (r<R)?
Maximum voltage of a Van de Graaff generator.
• Molecules in air get ionized for electric fields greater than roughly Emax = 3 x 106 V/m. What is the maximum voltage of a charged sphere of radius R=0.2 m?
Clicker Question Two identical positive charges of charge Q are a distance d
apart. What is the voltage at the midway point between the charges?
a) k Q/d b) 2 k Q/d c) 4 k Q/d d) 8 k Q/d e) 0
Clicker question Location P is equidistant from the two charges of an electric
dipole. The voltage at P is
a) positive b) zero c) negative
Electric potential of a charged ring • For a uniformly charged ring of
total charge Q, integration gives the potential on the ring axis:
• Very hard integral in general! If P is on x axis, then r is independent of θ.
V =�
k dq
rdq = λadθ
V (x, y, z) =� 2π
0
kλa dθ
r(θ, x, y, z)
• Integrating the potentials of charged rings gives the potential of a uniformly charged disk:
• This result reduces to the infinite-sheet potential close to the disk, and the point-charge potential far from the disk.
V (x) =�
kdQ
r=
� a
0k
λ2πr dr√x2 + r2
Equipotentials • An equipotential is a surface on which the potential is
constant. • In two-dimensional drawings, we
represent equipotentials by curves similar to the contours of height on a map.
• The electric field is always perpendicular to the equipotentials. (∆V = − �E · ∆�s = 0)
Conductors • There’s no electric field inside a conductor in
electrostatic equilibrium. • And even at the surface there’s no field
component parallel to the surface. • Therefore it takes no work to move charge
inside or on the surface of a conductor in electrostatic equilibrium.
A conductor in electrostatic equilibrium is an equipotential. • The electric field must be perpendicular to the
surface of a conductor (in electrostatic equilibrium
• For a very small displacement, • Suppose
Then
Can do the same thing in other direction:
The derivatives here are partial derivatives, expressing the variation with respect to one variable alone.
Determining E from V? • Voltage can be determined if electric field is known • Can electric field be determined if voltage is known?
∆V = − �E · ∆�r
�E · ∆�r = Ex ∆x
Ex = −∆V
∆x= −∂V
∂x
�E = −∇V = −�
∂V
∂xi +
∂V
∂yj +
∂V
∂zk
�
(gradient of V)
∆�r = ∆x i
• For which region is the magnitude of the electric field the highest?
80 V
Distance (cm)1 2 3 4 5 6 7 8 9 10
Dist
ance
(cm
)
4
65
7
8
9
3
2
1
A
B
200 V180 V
160 V
140 V120 V
D
C100 V
1. A 2. B 3. C 4. D