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Can electrons not flow through a load if they do not have a path to flow to the positive terminal of the battery? Why is it absolutely necessary for electrons to end up at the positive terminal?

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It is not that it is just necessary. Electron flow, in the first place, OCCURS due to the potential difference between the 2 terminals - negative and positive terminals.

Emf or potential difference is the driving force of the electrons in the electric circuit which causes the electrons to flow from the negative terminal to the positive terminal.

If the circuit is not closed, electrons will not flow.

EDIT: As per the comment by brucesmitherson ;
In an open circuit, the driving force is not absent, it is just that it is not strong enough to make electrons jump outside of the metal, which requires a lot of energy.

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    $\begingroup$ the driving force is not absent, it is just that it is not strong enough to make electrons jump outside of the metal, which requires a lot of energy. Think about lightning bolts. $\endgroup$
    – user83548
    Commented Nov 13, 2015 at 18:54
  • $\begingroup$ @brucesmitherson Yes I agree with your view. I have updated my answer. $\endgroup$ Commented Nov 13, 2015 at 18:57
  • $\begingroup$ Why the downvote? Please explain.. $\endgroup$ Commented Nov 22, 2015 at 11:25
  • $\begingroup$ It was not me.. $\endgroup$
    – user83548
    Commented Nov 22, 2015 at 13:44
  • $\begingroup$ It seems quite plausibe; don't think it deserves downvote, IMO. +1. $\endgroup$
    – user36790
    Commented Nov 23, 2015 at 14:14

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