Why is it the integral from a to b, but not a to c?

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In summary, the integral is taken from a to b because that is where the electric field can be found in the region between the plates of the capacitor. The electric field is zero between b and c, so taking the integral from a to c would yield the same result. The integral is not a sum of potentials, but a line integral of the electric field.
  • #1
flyingpig
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Homework Statement



[PLAIN]http://img225.imageshack.us/img225/3342/unledyk.png [Broken]





The Attempt at a Solution



The solution is posted with the problem.

My question is that why is the integral from a to b and not a to c? Why are they summining the potential from a to b?
 
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  • #2
hello,

the region from a to b is the 'between the plates' part of the capacitor, so that is where the electric field can be found. both the sphere and shell are conductors so there will be no e-field inside those regions.

also, they are not 'summing the potential'; that is a line integral of the e-field over a path from the surface of the sphere to the inner surface of the shell. since the electric force is conservative, the result will be the same no matter what path is chosen.
 
  • #3
Then what's wrong if you take the integral from a to c?
 
  • #4
you can do that if you want, but the electric field between b and c is zero, so it will contribute nothing to the integral and you will get the same result.
 
  • #5


The integral from a to b is used because those are the limits of integration for the given function. The integral is a mathematical tool used to find the area under a curve and can be thought of as summing the function values over a given interval. In this case, the function is the potential and the interval is from a to b. The potential may not be defined or may be constant for values outside of this interval, so it is not necessary to integrate over a larger interval such as a to c. Additionally, the limits of integration are chosen based on the physical situation being modeled and may not necessarily correspond to the full range of the independent variable. Therefore, the integral is from a to b, as that is the relevant interval for the given problem.
 

1. Why is the integral from a to b instead of from a to c?

The integral is a mathematical concept that represents the area under a curve. The notation of "from a to b" indicates that the integration is being performed over a specific interval, in this case, from point a to point b. The use of different letters, such as c, simply represents a different interval and does not change the concept of integration.

2. Can't we just use any letters for the limits of integration?

While it is true that any letters can be used for the limits of integration, the choice of letters a and b is standard convention. It helps to differentiate the variables used in the function being integrated from the variables used to indicate the limits of integration.

3. What does the notation "dx" represent in an integral?

The "dx" in an integral represents an infinitely small change in the variable x. In other words, it indicates the variable with respect to which the function is being integrated.

4. Can the limits of integration be variables instead of numbers?

Yes, the limits of integration can be variables. In fact, this is often the case when calculating definite integrals, where the upper and lower limits are not specified. In this case, the result of the integration will be a function of the variable used for the limits.

5. Why do we use integrals in scientific calculations?

Integrals are used in scientific calculations because they allow us to find the area under a curve, which is necessary for many real-life applications. For example, integrals are used to calculate the volume of irregularly shaped objects, the work done by a variable force, and the average value of a function over an interval.

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