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When the switch is closed, the current I2 = 100mA, the task is to find Pe1 before the switch is closed and after it's closed and compare them.

Now I know that $$\Delta P_{E_1}=P_{E_1}^{\left(2\right)}-P_{E_1}^{\left(1\right)}=E1\left(I_1^{\left(2\right)}-I_g\right)$$

I am not sure how to start the task, finding and Thevenin equivalent for the rest of the circuit in respect of the branch with the switch isn't possible as not enough information is given.. any hints, if it's hard to provide a complete solution I would like some tips/hints in the right direction, like what theorems should be applied to solve this.

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I would work out the problem with variables for the terms you don't know. It's possible that those terms cancel out. For example, Ig * E1 is a constant, and may cancel out for the change in power that you are trying to find. Besides, even if it's impossible because your teacher made a mistake, you can solve it in seconds once you get the missing information, if you already have the full equation.

Note that people won't give full solutions to homework problems on this forum, and won't really provide help, if you haven't shown a significant amount of effort in trying to solve it yourself.

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