In school we are taught about the motor effect and the dynamo effect. We are also taught Fleming's left-hand rule and Fleming's right-hand rule to describe the motor effect and dynamo effect (respectively).
We are taught that Fleming's left-hand rule describes the force that acts on a wire (technically the charge-carrying particles) that has a current which passes through a magnetic field.
We are taught that Fleming's right-hand rule describes the direction of a induced emf (or current if the circuit is complete) when a conductor is passed through a magnetic field.
The only difference between these that I can see is the difference between whether the force is inducing the current or if the current is inducing the force. To me, it would make sense that in either scenario the current would flow in the same direction, regardless of whether it is doing work on the system or if the system is doing work to produce it. However that's not how it works, if the current induces the force, the current is flowing in the opposite direction to which it would if the force induced the current.
I'm clearly misunderstanding something here, any help would be appreciated.