Is Work Done by External Forces Always Zero in Electrostatic Systems?

[saubhik]

I am stuck with some misconceptions about electrostatic potential/ potential difference. Please read what follows and correct me wherever I am wrong. Help appreciated. Thanks.



Consider a dipole-like charge configuration (fixed at their positions by unspecified forces ) with the separation between the two equal and opposite point charges $$2a$$. Clearly, there is a point between the charges where the electrostatic potential due to configuration is zero, assuming that the electrostatic potential due to the charge configuration at a point infinitely separated from the configuration is zero. In fact, this point say, $$P$$ is the mid point of the separation; the point being separated from each charge by $$a$$.

Now, the work done by external forces in bringing a point charge $$q$$ to the mentioned point infinitesimally slowly, from infinity, is zero since the electrostatic potential difference between the initial and final position of the charge $$q$$ is zero. Now, the work done in moving the point charge by conservative electrostatic forces is independent of the path taken which implies that the work done by the external forces being equal and opposite to the conservative electrostatic force is also independent of the path taken by the point charge. [Thus the external force behaves like a conservative force. In general, it can be said any force acting on a particle in a conservative force field, if displaces the particle infinitesimally slowly, is a conservative force.]

Suppose, the external agent brings the point charge $$q$$ from infinite separation (from the configuration) to the point $$P$$ through a path that goes through one of the fixed charges constituting the configuration. At the point at which the fixed charge is present, the electrostatic potential is not well defined. Therefore at a point in this specific path the electrostatic potential loses its definition. How can we ,then, say that the work done by the agent through this path is zero? (Assume that the electrostatic potential due to the charge configuration is zero at a point infinitely separated from the charge distribution.)

Suppose, the external agent now brings the point charge $$q$$ from infinite separation (from the configuration) to the point $$P$$ through a straight path that goes through the vicinity of one of the fixed charges. (Visualize this path as a straight line making a small angle with the dipole axis, extending infinitely at one direction and terminating at the point $$P$$). The electrostatic potential at each point in this path is contributed mostly by the fixed charge in the vicinity, since the distance of any point on this path (obviously, except the final point $$P$$) from the fixed charge in the vicinity is lesser than the distance from the other fixed charge. Thus, at each point ,except possibly the final point and initial point, in this path, the work done by the external agent is always non zero and has the same nature (always positive or always negative depending on the polarity of the fixed charge in vicinity of the path). How can the work done by the external force through this path be zero, which is the integration of the work done by external agent at each point of the path?

 

[chaoseverlasting]

First off, you won't be able to take the charged particle from infinity through one of the fixed charges. Secondly, when you bring the charged particle near the vicinity of one of the charges, the system changes and the potential is no longer zero so finite work is done.

The external force can take any path, but then some work needs to be done on the dipole itself so that its position remains constant.

Thus the external force behaves like a conservative force. In general, it can be said any force acting on a particle in a conservative force field, if displaces the particle infinitesimally slowly, is a conservative force.

can you justify this statement?

 

[saubhik]

Secondly, when you bring the charged particle near the vicinity of one of the charges, the system changes and the potential is no longer zero so finite work is done.

The charges of the configuration are fixed by unspecified forces, thus unable to move no matter how large a force acts on each one of them (so how is the system changing?). The charge q is brought to the point P through a path going through the vicinity of one of the fixed charges.Only the potential at the plane perpendicular to the dipole axis and containing the point P is zero ,apart from a point at infinity. The potential is non-zero at every other point in surrounding space. However, my question is that though the work done by agent at every proper point is non zero and has same nature at that specific path, how does the work done along that path sum to zero. [If I misunderstood your statement, kindly elaborate.]


In general, it can be said any force acting on a particle in a conservative force field, if it displaces the particle infinitesimally slowly parallel or antiparallel to field direction, is a conservative force.
Shouldn't it be? The external force (external to the system comprising of 3 charges) here is equal and opposite to a conservative force.

 

[saubhik]

Anyone? Please help.

 

[rwisz]

First of all, it is important to understand the concept of electrostatic potential and potential difference. Electrostatic potential is a measure of the energy required to bring a unit positive charge from infinity to a specific point in an electric field. It is a scalar quantity and is usually denoted by the symbol V. Potential difference, on the other hand, is the difference in electrostatic potential between two points in an electric field and is denoted by the symbol ΔV. It is also a scalar quantity and is measured in volts (V).

Now, in the case of a dipole-like charge configuration, the electrostatic potential due to the configuration is zero at a point infinitely separated from the configuration. This means that if we bring a unit positive charge from infinity to this point, no work is required. However, as we move closer to the configuration, the electrostatic potential increases due to the presence of the charges. At the midpoint between the two charges, the potential is zero because the contributions from the two charges cancel each other out.

In the first scenario mentioned, where the point charge is brought from infinity to the midpoint between the two charges, the work done by external forces is zero because the potential difference between the initial and final positions is zero. This is because the potential at the initial point (infinity) is zero and at the final point (midpoint), it is also zero. This is independent of the path taken because the potential at any point between the initial and final points is also zero.

In the second scenario, where the point charge is brought from infinity to the midpoint through a path that goes through one of the fixed charges, the potential at that specific point may not be well-defined. However, this does not affect the overall potential difference between the initial and final points, which is still zero. This is because the contributions from the two charges at the initial and final points cancel each other out, regardless of the potential at any other point in the path.

In the third scenario, where the point charge is brought from infinity to the midpoint through a straight path that goes through the vicinity of one of the fixed charges, the potential at each point may not be zero. However, the potential difference between the initial and final points is still zero because the contributions from the two charges at the initial and final points cancel each other out. The work done by the external force at each point may not be zero, but the overall work done is still zero because the potential difference is