Resistors in circuits

[lha08]

Homework Statement

I'm confused, like if there's a resistor in a closed circuit with a high resistance, does this mean that there is less current passing through it, and an increase in potential difference across this particular resistor and there is less power delivered?
And for a resistor with a low resistance, there is more current passing through and a decrease in potential difference across it and more power delivered?
Does this make sense? Is there anything else important that I should know about resistors?

Homework Equations

The Attempt at a Solution

 

[ehild]

All voltage sources have some internal resistance, and this has to be counted to the external resistance in the circuit.

You have a voltage source with emf E and internal resistance r, and you connect a resistor R making a closed circuit. The current flowing in this circuit is

$$I=\frac{E}{R+r}$$.

The potential difference across the resistor R is

$$V=IR=E\frac{R}{R+r}$$,

the higher R the lower the current and the higher the potential difference across R. If R varies from zero to infinity, the potential difference changes from zero to E and the current changes from E/r to zero.

The power delivered to the resistor is P=VI.

$$P=VI=\frac{E^2 R}{(r+R)^2}$$

It can be shown that the power has its highest value when r=R and

$$P_{max}=VI=\frac{E^2}{4r}$$

ehild