Calculating the electric potential

In summary, the conversation discusses the problem of finding the electric potential inside and outside of a metal pipe that has been split into four sections with different electric potentials. The equations for electric potential and boundary conditions are provided, and the conversation concludes with a discussion on how to solve for the coefficients and use the boundary conditions to find the values of ao and bo.
  • #1
physicsisfun0
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Homework Statement


upload_2018-11-13_21-27-41.png

We have the cross section of a metal pipe that has been split into four sections. Three of the sections have a constant electric potential, Vo. The fourth section is grounded so electric potential is zero. We are looking for electric potential inside and outside of the pipe.

Homework Equations


For electric potential I have:
V = ao + boln(s) +Σ(sv(avcos(vφ) + bosin(vφ)) + s-v(cvcos(vφ) + dvsin(vφ))
We also know boundary conditions:
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The Attempt at a Solution


(I think this next part is right)
And electric potential inside becomes:
V = ao + boln(s) +Σ(sv(avcos(vφ) + bosin(vφ))
And electric potential outside becomes:
V = ao + boln(s) +Σs-v(cvcos(vφ) + dvsin(vφ))

There is also symmetry along the x-axis so we can ignore the sin(vφ) contribution:
V = ao + boln(s) +Σ(sv(avcos(vφ))
V = ao + boln(s) +Σ(s-v(avcos(vφ))

Do I need to solve for ao and bo? How would I use the boundary conditions?
 

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  • #2
Welcome to PF!

physicsisfun0 said:
upload_2018-11-14_22-5-13.png
Note that the first equation above implies that ##7 \pi / 4## is less than ##\pi/4##. But, the intention is clear.

V = ao + boln(s) +Σ(sv(avcos(vφ))
V = ao + boln(s) +Σ(s-v(avcos(vφ))
OK, but it might be confusing to use the same notation for the coefficients for the two different regions.
Do I need to solve for ao and bo? How would I use the boundary conditions?
For the region inside the pipe, note that s can be zero. What does that tell you about the value of bo for the inside region?
For the region outside the pipe, note that s can become arbitrarily large. What does that tell you about bo for the outside region?

To find ao, set s equal to the radius of the pipe in the above two equations. For each equation, integrate both sides of the equation with respect to φ from 0 to 2π. The boundary condition tells you V as a function of φ for the integration of the left side.

To find other coefficients an, multiply both sides of the equation by cos(nφ) before integrating.
 

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  • Like
Likes Delta2

What is electric potential?

Electric potential, also known as voltage, is a measure of the amount of potential energy per unit charge at a specific point in an electric field.

How is electric potential calculated?

Electric potential can be calculated by dividing the work done in moving a charge from infinity to a specific point in an electric field by the amount of charge moved.

What is the unit for electric potential?

The unit for electric potential is volts (V), which is equivalent to joules per coulomb (J/C).

How does distance affect electric potential?

Distance plays a critical role in electric potential, as it follows an inverse-square law. This means that as distance increases, electric potential decreases at a rate of 1/r^2, where r is the distance between the point and the source of the electric field.

What is the difference between electric potential and electric potential energy?

Electric potential is a measure of the amount of potential energy per unit charge at a specific point, while electric potential energy is the total amount of potential energy possessed by a charge in an electric field. In other words, electric potential is a per-unit measure, while electric potential energy is a total measure.

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