Electrostatics and potential of points around charged plates

For an infinite plane charged surface electric field is constant above it no matter how far we go from the surface. So if the electric field remains constant in magnitude, how can the potential difference vary with distance . I am confuse as with distance if potential decreases electric field strength also decreases.and the force on another charge will also decrease if it goes it weaker areas of electric field strength. And in case of infinite sheet, EFL remains constant and hence the force but potential decreases with distance if the sheet is postive surface charged density.

2 Answers

The electric field $E$ is uniform, like that in a parallel plate capacitor. Because the plane is infinite in size it looks the same no matter how close to it or far from it you are. The electric field from the plate does not get weaker as you get further from the plane.

As is often the case when dealing with infinities, the absolute potential at any point is difficult to define. We usually define the potential as being zero at an infinite distance from the charged object, where the electric field is assumed to be zero. But in this case it never reaches zero even at infinity.

However, there is no problem in defining the potential difference between two points : it is simply $E$ x distance between the points. We could even define the potential to be zero at the plane - as we do when defining gravitational potential energy in a uniform field above a flat Earth, defining zero potential to be at ground level.

There is no conflict between a constant value of $E$ and a potential difference (increase or decrease) which varies with distance, just as in the gravitational case a constant value of $g$ gives a potential difference $gh$ per unit mass between two points a distance $h$ apart. $E$ is the force per unit charge, whereas potential $\phi$ is the energy per unit charge done to move the charge through a distance $x$ against that force. The force can be constant while we do a variable amount of work depending how far we move.

• Alright this makes sense.But when can we find potential of a point to be constant.is it when we have no electric field lines or also in some other case Apr 18, 2017 at 10:19
• Sorry Akash, I am not sure I understand your question. The electric field in space is like the landscape. The potential is like height above sea level, and equipotential lines are like contours on a map, joining places of equal height. The electric field strength is like the slope of the land, 1:1 for very steep, 1:10 for a gentle hill. Apr 18, 2017 at 15:06
• ... The potential at any point (height above sea level) is always constant whether the electric field strength is zero (land perfectly flat) or very large (land very steep) - unless charges are moving about (earthquakes which change the landscape). Apr 18, 2017 at 15:09

The electrostatic potential difference between two points $\vec r_1$ and $\vec r_2$ is defined as $$\phi=-\int_{\vec r_1}^{\vec r_2} \vec Ed\vec r$$ Thus for a constant electric field $\vec E$ with absolute value $E$ and an integration path in field direction, the potential difference becomes $$\phi=-E ∣\vec r_2 - \vec r_1∣=-E d$$ Thus, in a constant homogeneous electric field of strength $E$, the potential varies linearly with distance $d$ between two points on the same field line.