Force on a point charge due to a rod

• stunner5000pt
In summary, the discussion focuses on calculating the force on a point charge q due to a uniformly charged rod of length L at a distance x from the point charge. The force is found to be dependent on the distance and the charge of the rod, but not on its length. When the length of the rod approaches infinity, the force approaches 0. This is similar to the electric force due to a half of an infinite cylinder on a point charge.

stunner5000pt

Hvae a look at the diagram
Caluclate the force on the point charge q, due to a uniformly charged rod of length L a distance x from the point charge q. Discuss the limit when L approaches infinity with lambda = Q/L fixed.

$$Q = \lambda L$$
$$dQ = \lambda dL$$

Force of dQ on the point charge q is given by
$$dF = \frac{1}{4 \pi \epsilon_{0}} \frac{qdQ}{(L+x-s)^2}$$
no Y components since the rod is thin. SO this force is the total force in the horizontal direction only.

$$F = \int dF = \int_{s=0}^{s=L} \frac{1}{4 \pi \epsilon_{0}} \frac{q \lambda ds}{(L+x-s)^2}$$

$$F = \frac{q \lambda L}{4 \pi \epsilon_{0}} \frac{1}{x(L+x)}$$

$$F = \frac{Qq}{4 \pi \epsilon_{0} x(L+x)}$$

now for the limit where L -> infinity

i used L'Hopital's Rule and got the answer to be zero. But i find it hard to believe that that is the case. I Would think that this has something to di wth the limtis of integration being s=0 to s=infinity

im not sure however... do help

Last edited:
The mistake is pretty simple: you forgot that $Q = \lambda L$ when you took the limit.

ok since lambda is fixed it turns into

$$\frac{q \lambda}{4 \pi \epsilon_{0} x}$$

now how do i interpret this? This is certainly not similar to the force due to any object on a point charge... or is it ??

could someone guide me to the end of the problem?

i found out htat with the inifnite length the force doesn't dpeend on the length of the rod. It only depends on the distance from the rod. This resembles the electric force due to an half of an infinite cylinder on a point charge q. Is this much of an explanation enough ?

what is the conclusion for the second part where the rod is infinitely long? The force or electric field certainly do not depend on its length. Is there anything else i may have missed?

1. What is the formula for calculating the force on a point charge due to a rod?

The formula for calculating the force on a point charge due to a rod is F = k * (q1 * q2) / r^2, where F is the force, k is the Coulomb's constant, q1 is the charge of the point charge, q2 is the charge of the rod, and r is the distance between the point charge and the rod.

2. How does the distance between the point charge and the rod affect the force?

The force between a point charge and a rod is inversely proportional to the square of the distance between them. This means that as the distance increases, the force decreases.

3. What is the direction of the force on a point charge due to a rod?

The direction of the force on a point charge due to a rod is along the line connecting the point charge and the center of the rod. If the point charge is positive, the force will be repulsive, and if the point charge is negative, the force will be attractive.

4. Can the charge on the rod affect the force on the point charge?

Yes, the charge on the rod can affect the force on the point charge. The force is directly proportional to the charge of the rod, meaning that as the charge on the rod increases, the force on the point charge also increases.

5. How does the length of the rod impact the force on the point charge?

The length of the rod does not directly impact the force on the point charge. However, if the charge on the rod is distributed non-uniformly, the force on the point charge may vary along the length of the rod. In general, a longer rod with a uniform charge distribution will have a stronger force on the point charge compared to a shorter rod with the same charge distribution.