Transcript Document

Ch 25.5 – Potential due to Continuous Distribution
• We have an expression for the E-potential of a point charge.
q
V  ke
r
Electric potential
due to a point
charge.
• Now, we want to find the E-potential due to weird distributions of
charge.
Ch 25.5 – Potential due to Continuous Distribution
• Approach is similar to that for calculating E-fields.
• Break down the overall distribution into little chunks, find the Epotential due to each, and add up the contributions to get
dq
dV  ke
r
dq
V  ke 
r
Electric potential
due to a continuous
charge distribution.
EG 25.5 The electric potential due to a ring
(a) Find an expression for the electric
potential at a point P located on the
central axis of a uniformly charged
ring of radius a and total charge Q.
(b) Find an expression for the magnitude
of the E-field at point P.
EG 25.6 The electric potential due to a disk
A uniformly charged disk has radius R
and surface charge density σ.
(a) Find the electric potential at a point
P located on the disk’s central axis.
(b) Find the x component of the E-field
at P along the central axis of the
disk.
EG 25.7 The electric potential due to a line
A rod of length l located on the x axis
has a total charge of Q and a
uniform linear charge density of
λ = Q/l. Find the E-potential at
point P located on the y axis a
distance a above the origin.
Hint:

dx
x a
2
2

 ln x  x 2  a 2

Ch 25.6 – E-Potential Due to a Charged Conductor
Already know:
1) If a conductor carries net charge, it resides on the surface.
2) The E-field just outside the conductor is perpendicular to the surface.
Now, we show:
Every point on the surface of a charged conductor in equilibrium is at
the same electric potential.
Ch 25.6 – E-Potential Due to a Charged Conductor
Consider two points, A and B, on the surface of a charged conductor.
Since the E-field is always perpendicular to the
surface, any path from A to B along the surface will
generate zero potential difference:
 
V  VB  VA   E  ds  0
B
A
Ch 25.6 – E-Potential Due to a Charged Conductor
The surface of any charged conductor in equilibrium is
an equipotential surface.
Because the E-field is zero inside the conductor, the
electric potential is constant everywhere inside the
conductor and equal to its value at the surface.
Ch 25.6 – E-Potential Due to a Charged Conductor
The electric potential and the electric field
due to a positively charged, spherical
conductor.
1) Charge is smeared evenly over the surface.
2) Electric potential is constant inside the
sphere and acts like the potential due to a
point charge outside the sphere.
3) The E-field is zero inside the sphere, and
acts like a positive point charge outside the
sphere.
Ch 25.6 – E-Potential Due to a Charged Conductor
Recall, surface charge density on a conductor is determined by the
radius of curvature.
Small radius of curvature (very pointy)  high surface charge
density.
Large radius of curvature (flat)  low surface charge density.
EG 25.8 Two connected charged spheres
Two spherical conductors of radii r1 and r2
are separated by a distance much greater
than the radius of either sphere. The
spheres are connected by a conducting
wire. The charges on the spheres in
equilibrium are q1 and q2, respectively.
Find the ratio of the magnitudes of the Efields at the surfaces of the spheres.
Ch 25.6 – Cavity within a conductor
Suppose a conductor contains a cavity.
Assume the cavity contains no charges.
The E-field inside the cavity must be
zero, even if an E-field exists
outside the conductor.
Ch 25.6 – Cavity within a conductor
Remember, every point on the
conductor is at the same potential.
The only way for the potential
difference to be zero from A to B is
if the E-field at all points inside the
cavity is zero.
 
V  VB  VA   E  ds  0
B
A
Must be zero.
Ch 25.6 – Cavity within a conductor
A cavity surrounded by conducting
walls is a field-free zone, as long as no
charges are inside the cavity.
Ch 25.6 – Corona discharge
Corona discharge often occurs near the surface of a charged
conductor, like a high voltage power line.
A high potential difference between the conductor and the
surrounding air can strip electrons from the air molecules.
The electrons accelerate, colliding with other air molecules. The
result is the emission of light.
Corona Discharge