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When calculating the electric field from a point above a line of charge using coulomb's law, the integral that comes up is of the form [itex]\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } [/itex]. But if the point we were asked for is right in the middle, the horizontal (cosine) components cancel out, leaving only the vertical components, so we'd have to throw in a sine term to make the expression correct. Why, if we add the sine and cosine components, the result is not equal to the expression without sine and cosines?

Namely,

[itex]\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } \neq { \left( \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } sin\theta \right) }^{ 2 }\quad +\quad { \left( \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } cos\theta \right) }^{ 2 } [/itex]

Namely,

[itex]\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } \neq { \left( \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } sin\theta \right) }^{ 2 }\quad +\quad { \left( \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } cos\theta \right) }^{ 2 } [/itex]

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