- #1
Zaphodx57x
- 30
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I am having a problem understanding this problem which references this exercise .
I tried it a couple different ways. I used
\begin{multline*}<br /> \Delta U(Potential Energy) = Uf - Ui = Uf - U(r = infinity) = Uf - 0 \\<br /> dU = Uf = -W = \int F * ds = \int E * Q * ds \\<br /> Uf = -\int E * Q * dr = -\int \frac{kqQ}{r^2} = \frac{kqQ}{r}<br /> \end{multline*}
I also tried using the bare potential energy equation and using two arbitrary surface areas (dA) on opposite sides of the sphere with a distance between charges of 2r , then integrating each over just half of the sphere, which provided the same result.
The problem I have is that this result gives me a pretty decent result for the mass of an electron if I assume that q = Q = charge of an electron. But this assumption seems to say that the electron is pushing against itself by its full charge.
That is a force \frac{QQ}{r^2} is present instead of a \frac {(Q)(q}{r^2} where q = \frac {Q}{n} or some other fractional or modified charge.
This almost makes sense to me in that if a sphere were to have a charge spread evenly over its surface and its radius is infinity, it still acts as if it has all its charge at its center and therefore would resist a decreasing radius of like charge.
If anyone can find a way to explain why this problem makes sense I would really appreciate it. I just don't understand how the Potential energy can contain the electron's charge twice, when there is only one charge to begin with.
I tried it a couple different ways. I used
\begin{multline*}<br /> \Delta U(Potential Energy) = Uf - Ui = Uf - U(r = infinity) = Uf - 0 \\<br /> dU = Uf = -W = \int F * ds = \int E * Q * ds \\<br /> Uf = -\int E * Q * dr = -\int \frac{kqQ}{r^2} = \frac{kqQ}{r}<br /> \end{multline*}
I also tried using the bare potential energy equation and using two arbitrary surface areas (dA) on opposite sides of the sphere with a distance between charges of 2r , then integrating each over just half of the sphere, which provided the same result.
The problem I have is that this result gives me a pretty decent result for the mass of an electron if I assume that q = Q = charge of an electron. But this assumption seems to say that the electron is pushing against itself by its full charge.
That is a force \frac{QQ}{r^2} is present instead of a \frac {(Q)(q}{r^2} where q = \frac {Q}{n} or some other fractional or modified charge.
This almost makes sense to me in that if a sphere were to have a charge spread evenly over its surface and its radius is infinity, it still acts as if it has all its charge at its center and therefore would resist a decreasing radius of like charge.
If anyone can find a way to explain why this problem makes sense I would really appreciate it. I just don't understand how the Potential energy can contain the electron's charge twice, when there is only one charge to begin with.
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