Energy Dissipation Across a Resistor

[boredbluejay]

Homework Statement

A battery is connected across a uniform resistor R₀. A sliding contact can move across R₀ from x=0 to x=10cm at the right. Moving the contact changes how much resistance R is to the left of the contact and how much is to the right. Find the rate at which energy is dissipated in R as a function of x.

Homework Equations

Energy dissipated: P_R=i²R
Voltage is the same for resistors in parallel
1/R_total=1/R+1/R₀
i_total=V/R_total

The Attempt at a Solution

I tried to express R in terms of x and R₀, but I got stuck. I honestly have no idea how to begin, or even if all of the equations that I wrote above are relevant to this question. Could someone please give me a hint that'll point me in the right direction? Thanks!

 

[tiny-tim]

hi boredbluejay!

isn't resistance proportional to length?

 

[boredbluejay]

I guess, but by what factor. >< I'm sorry, I'm really bad at this.

 

[lewando]

You have two points on the x,R plane: (0,0) and (10,R₀). With this, determine the linear relationship R = mx + b.

 

[Nickie]

Firstly, it is important to note that the sliding contact changes the resistance in the circuit by essentially creating two resistors in parallel. The resistance to the left of the contact (R1) can be expressed as R1 = R0(x/10) and the resistance to the right of the contact (R2) can be expressed as R2 = R0(1-x/10). This is because as the contact moves, the length of R1 and R2 changes in proportion to the total length of R0 (10cm).

Next, we can use the equations for power and current to find the rate at which energy is dissipated in R. The power dissipated in a resistor is given by P = i^2R, where i is the current flowing through the resistor. Since the voltage is the same for resistors in parallel, we can use the equation itotal = V/Rtotal to find the total current in the circuit.

Using this information, we can now express the power dissipated in R as a function of x:

P = (itotal)^2R = (V/Rtotal)^2R = (V^2/Rtotal)(R1||R2)

Where R1||R2 is the equivalent resistance of R1 and R2 in parallel. This can be found using the equation 1/Rtotal = 1/R1 + 1/R2.

Substituting in the expressions for R1 and R2, we get:

P = (V^2/Rtotal)(R0(x/10)(1-x/10))/(R0(x/10)+(1-x/10))

Simplifying, we get:

P = (V^2/R0)(x/10)(1-x/10)

Therefore, the rate at which energy is dissipated in R as a function of x is given by P = (V^2/R0)(x/10)(1-x/10). This shows that the power dissipated in R is dependent on both the voltage and the length of the resistor on either side of the contact. As the contact moves closer to one side, the length of that resistor increases and the length of the other resistor decreases, resulting in a change in the power dissipated.