Question

"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by
Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),
where k is a physical constant and a > 0.
a. Confirm that Eₓ(a)=kQ / a √(a²+L²)
b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.

Accepted Answer

Start with the given expression for the x-component of the electric field:  
$$E_x(a) = \frac{kQa}{2L} \int_{-L}^{L} \frac{dy}{(a^2$ + y^2$)^{3/2}$$}$$. Recognize that the integral is an even function because the integrand depends on y^2$. Therefore, rewrite the integral as:  
$$\int_{-L}^{L} \frac{dy}{$$(a^2$$ + $$y^2$$)^{3/2}$$} = 2 \$$int_0$$^{L} \frac{dy}{(a^2$ + y^2$)^{3/2}$$}$$. Use the substitution y = a$$ 	an 	heta$$, which implies dy = a$$ \$$sec^2$$ 	heta d	heta$$. Also, note that a^2$ + y^2$ = a^2$ \sec^2$ 	heta. $$Substitute these into the integral to transform it into an integral in terms of $$	heta. $$Simplify the integral after substitution:  
$$\$$int_0$$^{L} \frac{dy}{(a^2$ + y^2$)^{3/2}$$} = \$$int_0$$^{\arctan(L/a)} \frac{a \$$sec^2$$ 	heta d	heta}{$$(a^2$$ \$$sec^2$$ \$$theta)^{3/2}$$} = \$$int_0$$^{\arctan(L/a)} \frac{a \$$sec^2$$ 	heta d	heta}{$$a^3$$ \$$sec^3$$ 	heta} = \$$int_0$$^{\arctan(L/a)} \frac{\$$sec^2$$ 	heta}{$$a^2$$ \$$sec^3$$ 	heta} d	heta = \frac{1}{$$a^2$$} \$$int_0$$^{\arctan(L/a)} \cos 	heta d	heta$$. Evaluate the integral $$\$$int_0$$^{\arctan(L/a)} \cos 	heta d	heta = \sin 	heta \big|_$$0^{\arctan(L/a)}$$ = \sin(\arctan(L/a))$$. Use the right triangle relationship to express $$\sin(\arctan(L/a)) = \frac{L}{\sqrt{$$a^2$$ + $$L^2$$}}$$. Substitute back to get the integral value and then multiply by the constants outside the integral to confirm that  
E_x(a$$) = \frac{kQ}{a \sqrt{$$a^2$$ + $$L^2$$}}$$.

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Key Concepts

Electric Field Due to a Continuous Charge Distribution

Definite Integral and Substitution in Calculus

Limit Process and Charge Density Concept