Why Does the Electric Field Component Ex Require a Negative Sign?

AI Thread Summary
The discussion centers on understanding why the x-component of the electric field, Ex, requires a negative sign in calculations. The negative sign arises because the contributions from positive charge elements on one side of a vertical line outweigh those from the other side, leading to a net negative x-component when λ is positive. Participants emphasize that the final expression for Ex should not contain the variable x but rather be defined by the bounds of integration. The importance of applying the correct bounds to clarify the sign of the electric field is also highlighted. Overall, the negative sign reflects the directional nature of the electric field based on charge distribution.
yesmale4
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Homework Statement
A uniformly charged rod is positioned along the x-axis as shown in the figure. Calculate the electric field at a point P on the y-axis.
Relevant Equations
dq=lamda*dx
de=kdq/r^2
hello i would like to understand to something.
here is the drew
a.png

now for my question:
i was able to find Ey and here is my correct answer:
aa.jpg

when i try to find Ex i didnt understand something, i found the correct answer but i need to put minus before and i want to know why?
here is my solution for Ex (the correct answer):
aaa.jpg


yes i know i didnt solve the integral becuase first i want to know why there is minus
 
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There is more charge to the right of the vertical line than to the left. The sum of all the contributions from the charge on the right will be grater than the contributions from the charge on the right. So if λ is positive the overall x-component will be negative and the opposite if λ is negative.

Of course, this should be built in the answer. You say that your answer is correct, but how do you know that? It looks like the ends of the rod are at ##x=-a## and ##x=\delta,## Your final answer should have no ##x## in it; only ##a## and ##\delta## and be the sum of two expressions resulting from the integral.

Also, please use LaTeX. Your handwriting is hard to decipher at times.
 
yesmale4 said:
when i try to find Ex i didnt understand something, i found the correct answer but i need to put minus before and i want to know why?
What is the sign of the x-component of the arrow you drew?
 
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yesmale4 said:
why there is minus
Because the sign of the x component of the field due to an element dx at x has the sign of ##-\lambda x.dx##. I.e. if ##\lambda## is positive then where x is negative the field is to the right and vice versa.
kuruman said:
Your final answer should have no x in it
It doesn't really have an x in it - that's just a dummy variable. The OP has merely not done the final step of applying the bounds. Maybe that's what you meant.
And I think it is b, not ##\delta##.
 
haruspex said:
It doesn't really have an x in it - that's just a dummy variable. The OP has merely not done the final step of applying the bounds. Maybe that's what you meant.
Yes, that's what I meant. Applying the bounds would yield the difference of two terms in which case it would be easier to point out mathematically that the sign of the x-component depends on both the sign of the linear charge density and the relative size of the bounds.
 
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