# How can we hold a charge to be stationary in an electric field [closed]

Suppose we have a positive point charge And we put a positive test charge in in the electric field, how can we keep this charge static, And how external work May keep it static because i think it will continue to repel

## closed as unclear what you're asking by Chris♦, ZeroTheHero, user191954, Frobenius, John RennieSep 11 '18 at 9:48

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• You can't keep any charges static in a single field without any other interactions keeping it in place as well. Just like how to can't have equilibrium with just a single force. Why are you wanting to keep the charge stationary? – Aaron Stevens Sep 10 '18 at 20:54
• This is what Millikan did during his oil drop experiment. practicalphysics.org/millikan-experiment.html – Farcher Sep 10 '18 at 21:06
• There is by definition no work involved in keeping an object in a fixed position as is regularly pointed out on this site. – my2cts Sep 10 '18 at 21:29
• I want to keep it in equilibrium because i want to place it at a certain point. – Ahmad Eldesokey Sep 10 '18 at 21:29
• So you want to keep point charges stable in just some system? Or specifically in a system with just one other positive charge? – Aaron Stevens Sep 10 '18 at 21:37

There's a mathematical result called Earnshaw's Theorem that proves that keeping a point charge stably confined in an electrostatic field alone is impossible (specifically, it proves that there are no local maxima/minima in the electric potential in free space, only saddle points). If you use both an electrostatic and a magnetic field, however, confinement to a particular region is possible. In fact, this is the basis of the Penning trap, which is used in many laboratory situations to trap ions (see diagram below, from https://en.wikipedia.org/wiki/Penning_trap#/media/File:Penning_Trap.jpg). Since the uniform magnetic field makes charges travel in a helix, it confines the point charge trajectories to a thin cylinder, and the electric fields deflect the charges away from the ends of that cylinder.

In this context, I take "static" mean to zero velocity and acceleration. Zero velocity is a question of the choosing the right reference frame. Zero acceleration means zero net force, so you will have to apply a force to the test charge that is equal and opposite the force applied by the electric field. If the only (non-negligible) force acting on the test charge is the one applied by the electric field, then the charge will accelerate and move. In the situation you describe, it will be repelled by the positive point charge.

We usually neglect gravity in these scenarios, for a variety of reasons. But we can easily imagine a situation in which the attractive gravitational force on the test charge (due to the other particle) exactly balances the repulsive force applied by the electric field, yielding a zero net force.

You mentioned "external work." The work done by the net force acting on an object will equal that object's change in kinetic energy. If the object is static, the work done by the net force must be zero.

You can't have a stationary test charge if it only experiences the field of one other charge. It is impossible. As others have pointed out, if you include gravity or some other force than it is possible. But only using a single electric field, you cannot keep some charge in this field stationary.

This can be seen using Newton's second law. The magnitude of the net force experienced by the test charge is $$\sum F=F_1=q_{test}\cdot E=m_{test}\cdot a\neq0$$

Therefore, the test charge will accelerate. If you want $0$ acceleration, then you need some other force equal and opposite to the force produced by the field of the positive point charge $$\sum F=F_1+F_2=0$$ so that $$F_2=-q_{test}\cdot E$$

This force can be anything. Another electric field. Gravity. As long as the forces balance out you can get equilibrium, but you cannot do it with just a single force.