Why Does the Resistance Need to Match for Max Energy Transfer?

[Psi-String]

I was wondering why "for maximum transfer of energy from an emf device to a resistive load, the resistance of the emf device must equal the resistance of the load"?
So far, I don't have any idea. Could someone explain this to me? Thanks in advance.

 

[Psi-String]

Could someone give me a clue? I was thinking that maybe it's because when the resistance of the emf device and load is the same, the thermal dissipation of the emf device is reletively smaller. But I don't know whether this idea is right or not.

 

[berkeman]

I'm not sure that there is an intuitive explanation for it. I've always just written the equation for the power transferred to the load (with the output resistance of the source as a variable), and solved it for maximum power transferred to the load. Do that and you will see that Rout and Rload are equal for maximum power transfer to the load.

 

[Psi-String]

Well... this is my calculation, but it seems a little bit weird.
From loop rule

$$\epsilon =ir + iR$$

so $$i=\frac{\epsilon}{r+R}$$

thus the power transfer to the load is

$$i^2R =\frac{{\epsilon}^2 R}{(r+R)^2}$$
(where $$\epsilon$$ is emf, r is the resistance of the emf device which is a variable, R is the resistance of the load which is constant)
but it turns out that when r is 0 then the power transter to the load is maximum. It is reasonable but not the result I set for. Where did I do wrong?

 

[berkeman]

The two constants in the calculation will be the source voltage Vs and the source resistance Rs. The variable that you control to get maximum power transfer is R of the load. If Rload is very large, then you get all of Vs across the Rload, but very little current flows because Rload is so big. If Rload is very small, then you get maximum current out of the source, but very little voltage across the Rload. So the optimium power transfer P=V*I is somewhere between Rload being big and small.

To find out what the optimum value of Rload is, write the equation for the power across Rload as a function of Rload, and use differentiation to maximize that power. When you do this, you should get Rload=Rs.

 

[Psi-String]

I got it. Thanks for help