Deriving Electric Field from Voltage

[Gear300]

This is probably more Calculus than it is physics. The voltage at a point X produced by a uniformly charged rod is V = (Ke*Q/l)*[ln(l+sqr(l^2+a^2))-ln(a)], in which point X is right above the left end of the rod by a distance a, l is the rod's length, Q is the charge of the rod, and Ke as the constant (sqr( ) refers to square root and ln is natural log).

X(point X a distance a from the left end of the rod, in which a is constant)


__________________ (uniformly charged rod)

I'm supposed to find the y-component of the Electric Field at X, in which I would just find the negative partial derivative of V in respect to y. Would that imply that I derive in respect to a? I've actually tried quite a number of derivations, but they always end up in something lengthy. The answer is Ey = Ke*Q/[a*sqr(l^2+a^2)].

 

[Shooting Star]

Would that imply that I derive in respect to a? I've actually tried quite a number of derivations, but they always end up in something lengthy. The answer is Ey = Ke*Q/[a*sqr(l^2+a^2)].

You are correct and so is the final answer. Try it once more. Remember, you have to differentiate wrt 'a' only. (There's a factor (l + sqrt(l^2+a^2) which cancels out.)

 

[Moetasim]

I would like to clarify that the process of deriving the electric field from the voltage is indeed a fundamental principle in physics, specifically in the field of electromagnetism. While it may involve some concepts from calculus, the underlying physics principles are still at play.

The equation provided is the mathematical representation of the voltage at a point X produced by a uniformly charged rod. In this case, the voltage is a function of the distance from the left end of the rod (a) and the length of the rod (l), as well as the constant (Ke) and the charge of the rod (Q).

To find the y-component of the electric field at point X, you would indeed need to take the negative partial derivative of the voltage (V) with respect to y. This implies that you would need to differentiate the equation with respect to a, since a is the variable that represents the distance in the y-direction.

The resulting equation for the y-component of the electric field (Ey) is Ke*Q/[a*sqr(l^2+a^2)]. This result may seem lengthy, but it accurately represents the relationship between the voltage and the electric field at point X.

In conclusion, the process of deriving the electric field from the voltage is a valid and important aspect of physics, and while it may involve some calculus, the underlying principles are still rooted in physics. It is essential to understand and apply these principles in order to fully comprehend the behavior of electric fields and their effects on charged particles.