Charging by Induction: How to explain it? I am teaching high school physics and our textbook says that "If a charged rod is brought close to the electroscope and the cap is earthed momentarily, the electroscope is left with the opposite  to that on the rod." 
And the usual explanations for it goes as this:
If a negatively charged rod is brought close to the cap of the electroscope, electrons are repelled to the leaves, leaving the cap positive. If the cap is earthed, by a finger say, the electrons run away to earth. If the finger is removed  first, followed by the rod, the electroscope is left positively charged. 
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The question asked: How do the electrons overcome the repulsion of the negative rod to run to earth?
 A: The negative charges will want to get as far away from each other as possible. Since our bodies are fairly good conductors, this is fairly easy for the negative charges to move towards ground to achieve this. The repulsion of the negative charges on the rod is just not enough to completely contain all of the negative charge in the electroscope. Negative charges in the tip are much closer to each other than they are to the rod, so this repulsion will win initially to create an overall net positive charge on the electroscope.
If this is unconvincing you could go this route instead. To get a net positive charge on the electroscope we do not need to remove much negative charge from the electroscope. Even though the diagram shows positive charge at the tip before we touch it, this does not mean it is devoid of negative charge. It just means there is a net positive charge there. So when we touch the tip there will be some negative charges where the negative rod will actually aid in pushing these charges to ground through the finger. This is enough to give the electroscope a net positive charge since some negative charge has left it.
A: There are three charged areas to consider here. The negatively charged rod, the positively charged cap at the top of the electroscope and the negatively charged leaves. What happens when you use the superposition principle and determine the combined electric field?
You should find that the field between the cap and the leaves points towards the leaves and therefore allows electrons to move upwards.
A: The distribution of electrons in the electroscope before the cap is grounded has no bearing on the behavior of electrons when the cap is grounded, so we might as well just consider what the electrons would do if the cap was grounded before the negatively charged rod was brought in.
In this scenario, the electrons in the cap, repelled by the rod, have two paths of escape: tiny leaves with tiny capacitance and the earth with practically infinite capacitance. Obviously, electrons will flow to the ground until the potential on the cap becomes zero. 
If we did consider this as a two-step process, we could observe that the potential of the leaves (and the cap) after the rod is brought in would be negative. As soon as the cap is grounded, its potential becomes zero, overpowering the influence of the rod and attracting excessive electrons in the leaves.         
