Deriving Electric Field from Voltage

In summary, The conversation discusses finding the y-component of the electric field at a point X, which is right above the left end of a uniformly charged rod. The formula for calculating the voltage at point X is V = (Ke*Q/l)*[ln(l+sqr(l^2+a^2))-ln(a)]. The final answer for the y-component is Ey = Ke*Q/[a*sqr(l^2+a^2)], which is found by taking the negative partial derivative of V with respect to y and differentiating with respect to a. The factor (l + sqrt(l^2+a^2) cancels out during the process.
  • #1
Gear300
1,213
9
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)].
 
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  • #2
Gear300 said:
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.)
 
  • #3


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.
 

Related to Deriving Electric Field from Voltage

1. What is the relationship between electric field and voltage?

The electric field and voltage are directly proportional to each other. This means that as the voltage increases, the electric field also increases. Similarly, as the voltage decreases, the electric field decreases.

2. How do you calculate electric field from voltage?

The formula for calculating electric field from voltage is E = V/d, where E is the electric field in volts per meter (V/m), V is the voltage in volts (V), and d is the distance between the two points in meters (m).

3. Can electric field be negative?

Yes, electric field can be negative. This means that the direction of the electric field is opposite to the direction of the positive charges. Positive electric field indicates a repulsive force, while negative electric field indicates an attractive force.

4. How does the direction of the electric field relate to the direction of the voltage?

The direction of the electric field is always perpendicular to the equipotential lines (lines of equal voltage). This means that the direction of the electric field is always perpendicular to the direction of the voltage.

5. What are the units of electric field and voltage?

The units of electric field are volts per meter (V/m), while the units of voltage are volts (V). Both units are derived from the SI unit of electric potential, which is the volt (V).

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