- #1
PhysicsKush
- 29
- 4
- Homework Statement
- Given a two-conductor setup of two infinitely long parallel wires. Let the conducting wires have radius ##R## and be separated by a distance ##D##, where ##D## is much larger than ##R##. Use the approximation that the field from the second wire does not impact the charge distribution on the first. Similarly, you may assume that the field near a wire is dominated by that wire and you can safely neglect the field from the other wire, and that the wire diameter is small compared to the separation. Where then is the electric field going to be maximum ? What is the potential difference between the two wires, expressed in terms of ##\lambda## ?
- Relevant Equations
- For a wire
$$V = \frac{\lambda}{2 \pi \epsilon_{0}} \ln \Bigg\lvert \frac{r}{a} \Bigg\rvert,$$
where ##r## is the distance to the point and ##a## is the distance from the reference point.
$$ E = \frac{\lambda}{2 \pi \epsilon_{0} r},$$
where ##r## is the radius of the wire.
For the first part, since
$$ E(r) \propto \frac{1}{r} \hat{r}$$
by the principle of superposition the maximal electric field should be halfway in between the two wires.
Then I'm not sure how to go about the second part of the question. I understand that the total potential due to the two wires on an ##2## dimensional plane , at an arbitrary point is
$$ V = \frac{\lambda}{2 \pi \epsilon_{0}} \ln\Bigg\lvert\frac{d_{-}}{d_{+}}\Bigg\rvert,$$
where ##d_{-} ,d_{+}## are respectively the distances between each wire to the evaluated point. I also know that the potential difference between two points is
$$ V(b) - V(a) = \int_{a}^{b} \vec{E} \cdot d\vec{l},$$
but I'm unable to translate these informations into a correct answer.
Any insight would be appreciated
$$ E(r) \propto \frac{1}{r} \hat{r}$$
by the principle of superposition the maximal electric field should be halfway in between the two wires.
Then I'm not sure how to go about the second part of the question. I understand that the total potential due to the two wires on an ##2## dimensional plane , at an arbitrary point is
$$ V = \frac{\lambda}{2 \pi \epsilon_{0}} \ln\Bigg\lvert\frac{d_{-}}{d_{+}}\Bigg\rvert,$$
where ##d_{-} ,d_{+}## are respectively the distances between each wire to the evaluated point. I also know that the potential difference between two points is
$$ V(b) - V(a) = \int_{a}^{b} \vec{E} \cdot d\vec{l},$$
but I'm unable to translate these informations into a correct answer.
Any insight would be appreciated