A uniform linear charge of 2.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the x component of the electric field at y = 2 m on the y axis?
A) $$\int _ { 0 } ^ { 3 } \frac { - 18 x d x } { \left( 4 + x ^ { 2 } \right) ^ { 3 / 2 } }$$
B) $\int _ { 0 } ^ { 3 } \frac { - 18 d x } { ( 2 + x ) ^ { 2 } }$
C) $\int _ { 0 } ^ { 3 } \frac { - 18 d x } { 4 + x ^ { 2 } }$
D) $\int _ { 0 } ^ { 3 } \frac { - 18 x d x } { ( 2 + x ) ^ { 3 } }$
E)none of these

Question

Quizplus · Accepted Answer

The correct integral is:$$\int _ { 0 } ^ { 3 } \frac { - 18 x d x } { \left( 4 + x ^ { 2 } ight) ^ { 3 / 2 } }$$Explanation:The x component of the electric field due to a line charge is given by:$$E_x = \int \frac { \lambda dx } { 4 \pi \epsilon_0 r^2 }$$where:$$\lambda$$ is the linear charge density$$\epsilon_0$$ is the permittivity of free space$$r$$ is the distance from the line charge to the point where the electric field is being calculatedIn this case, the line charge is distributed along the x axis from x = 0 to x = 3 m, and the point where the electric field is being calculated is at y = 2 m on the y axis. Therefore, the distance from the line charge to the point where the electric field is being calculated is:$$r = \sqrt { x^2 + y^2 } = \sqrt { x^2 + 4 }$$Substituting this into the equation for the x component of the electric field, we get:$$E_x = \int _ { 0 } ^ { 3 } \frac { \lambda dx } { 4 \pi \epsilon_0 \left( x^2 + 4 ight) ^ { 3 / 2 } }$$Substituting the given values for $$\lambda$$ and $$\epsilon_0$$, we get:$$E_x = \int _ { 0 } ^ { 3 } \frac { 2 	imes 10 ^ { - 9 } dx } { 4 \pi \left( 8.85 	imes 10 ^ { - 12 } ight) \left( x^2 + 4 ight) ^ { 3 / 2 } }$$Simplifying, we get:$$E_x = \int _ { 0 } ^ { 3 } \frac { - 18 x dx } { \left( 4 + x ^ { 2 } ight) ^ { 3 / 2 } }$$Therefore, the correct integral is A.

A Uniform Linear Charge of 2 ∫03−18xdx(4+x2)3/2\int _ { 0 } ^ { 3 } \frac { - 18 x d x } { \left( 4 + x ^ { 2 } \right) ^ { 3 / 2 } }∫03​(4+x2)3/2−18xdx​

A Uniform Linear Charge of 2 $\int _ { 0 } ^ { 3 } \frac { - 18 x d x } { \left( 4 + x ^ { 2 } \right) ^ { 3 / 2 } }$