A rod of length L lies along the x-axis with its left end at the origin. The rod has a non-uniform charge density λ = ax, where a is a positive constant. (a) Express the total charge on the rod in terms of a and Z. (b) Calculate the electric field at point P, shown in the Figure. Take the limitd > L. What does the electric field look like in this limit? Is this what you expect? Explain. Hint: the following integral may be useful: a x dx = +In(x+a) (x+a)² x+a x² In(1+x)=x-+ x 2 (for small .x)

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A rod of length L lies along the x-axis with its left end at the origin. The rod has a non-uniform
charge density λ = ax, where a is a positive constant.
P
(a) Express the total charge Q on the rod in terms of a and L.
(b) Calculate the electric field at point P, shown in the Figure. Take the limit d » L. What does
the electric field look like in this limit? Is this what you expect? Explain. Hint: the following
integral may be useful:
x dx
(x+a)²
In(1+x)=x-
a -+ln(x + a)
X
x + a
+²
2
+... (for small .x)
Transcribed Image Text:A rod of length L lies along the x-axis with its left end at the origin. The rod has a non-uniform charge density λ = ax, where a is a positive constant. P (a) Express the total charge Q on the rod in terms of a and L. (b) Calculate the electric field at point P, shown in the Figure. Take the limit d » L. What does the electric field look like in this limit? Is this what you expect? Explain. Hint: the following integral may be useful: x dx (x+a)² In(1+x)=x- a -+ln(x + a) X x + a +² 2 +... (for small .x)
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