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A very thin plastic rod of length is rubbed with cloth and becomes uniformly charged with a total charge of . Consider an arbitrary piece of rod of length located at a position on the rod. Find an expression for the electric field due to an arbitrary source location at observation location on the x-axis, a distance to the right of the end of the rod as indicated. Next, integrate to find the total electric field caused at by the entire rod. Use for Coulomb's constant

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The total electric field of the rod at a distance, x to the right hand end of the rod is determined as [tex]E = \frac{\lambda}{4\pi \varepsilon _0} [\frac{1}{x} - \frac{1}{x+ l} ][/tex].

Expression for the electric field

An expression for the electric field due to an arbitrary source location at observation location on the x-axis is determined as follows;

E = kq/r²

where;

  • k is coulomb's constant
  • r is the distance on x - axis

Total electric field of the entire rod

[tex]\int\limits {dE} \,=\int\limits {\frac{kdQ}{r^2} }\\\\E = \lambda k \int\limits^{x + l}_x {\frac{dr}{r^2}} \,= \lambda k [-\frac{1}{r} ] \\\\E = \lambda k [\frac{1}{x} - \frac{1}{x+ l} ]\\\\E = \frac{\lambda}{4\pi \varepsilon _0} [\frac{1}{x} - \frac{1}{x+ l} ][/tex]

Thus, the total electric field of the rod at a distance, x to the right hand end of the rod is determined as [tex]E = \frac{\lambda}{4\pi \varepsilon _0} [\frac{1}{x} - \frac{1}{x+ l} ][/tex].

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