A ring has a uniform charge density
, with units of coulomb per unit meter of arc. Find the electric potential at a point on the axis passing through the center of the ring.
Strategy
We use the same procedure as for the charged wire. The difference here is that the charge is distributed on a circle. We divide the circle into infinitesimal elements shaped as arcs on the circle and use cylindrical coordinates shown in
[link] .
Solution
A general element of the arc between
and
is of length
and therefore contains a charge equal to
The element is at a distance of
from
P , and therefore the potential is
Significance
This result is expected because every element of the ring is at the same distance from point
P . The net potential at
P is that of the total charge placed at the common distance,
.
A disk of radius
R has a uniform charge density
, with units of coulomb meter squared. Find the electric potential at any point on the axis passing through the center of the disk.
Strategy
We divide the disk into ring-shaped cells, and make use of the result for a ring worked out in the previous example, then integrate over
r in addition to
. This is shown in
[link] .
Solution
An infinitesimal width cell between cylindrical coordinates
r and
shown in
[link] will be a ring of charges whose electric potential
at the field point has the following expression
where
The superposition of potential of all the infinitesimal rings that make up the disk gives the net potential at point
P . This is accomplished by integrating from
to
:
Significance
The basic procedure for a disk is to first integrate around
and then over
r . This has been demonstrated for uniform (constant) charge density. Often, the charge density will vary with
r , and then the last integral will give different results.
Find the electric potential due to an infinitely long uniformly charged wire.
Strategy
Since we have already worked out the potential of a finite wire of length
L in
[link] , we might wonder if taking
in our previous result will work:
However, this limit does not exist because the argument of the logarithm becomes [2/0] as
, so this way of finding
V of an infinite wire does not work. The reason for this problem may be traced to the fact that the charges are not localized in some space but continue to infinity in the direction of the wire. Hence, our (unspoken) assumption that zero potential must be an infinite distance from the wire is no longer valid.
To avoid this difficulty in calculating limits, let us use the definition of potential by integrating over the electric field from the previous section, and the value of the electric field from this charge configuration from the previous chapter.
Solution
We use the integral
where
R is a finite distance from the line of charge, as shown in
[link] .
With this setup, we use
and
to obtain
Now, if we define the reference potential
at
this simplifies to
Note that this form of the potential is quite usable; it is 0 at 1 m and is undefined at infinity, which is why we could not use the latter as a reference.
Significance
Although calculating potential directly can be quite convenient, we just found a system for which this strategy does not work well. In such cases, going back to the definition of potential in terms of the electric field may offer a way forward.
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?