Capacitance and electric potential

[scholio]

Homework Statement

a 100 pico Farad capacitor C_1 is charged to a potential difference of 50 volts. the charging battery is disconnected and the capacitor is then connected to a second uncharged capacitor C_2. if the potential across C_1 drops to 35 volts, what was the capacitance of C_2?

Homework Equations

capacitance C = Q/V where Q is charge, and V is electric potential in volts
units of capacitance is Farad

The Attempt at a Solution

let C_1 = Q/V_1 where C_1 = 100 pico farads = 100 *10^-12 farads and V_1 = 50 volts
solve for Q to get Q = 5*10^-9 coulombs

let C_2 = Q/V_2 where Q = 5*10^-9 coulombs and V_2 = 35 volts, solve for C_2 to get

capacitance of C_2 = 1.43*10^-10 Farads

did i do this correctly?

thanks

 

[G01]

Be careful! Does all the charge initially on the first capacitor move onto the second?

 

[scholio]

no you are right, all the charge of C_1 does not go to C_2. did soling the C_1 equation for q come into play for this type of problem?

how do i set up an equation that involves C_1 and C_2, better yet the change in capacitance of C_1?

or would the equation become C_2 = Q/ deltaV where Q = 5*10^-9 and V = 50 -35 = 25 so that C_2 = 2*10^-10 farads?

is this the correct approach?

 

[G01]

No, your right that C_2 has 35 V across it. Since the two capacitors are in parallel they must have the same voltage across them.

You know that after the battery is removed: Q_1+Q_2=5nC

Is there any way you can find Q_1 and then find Q_2?

HINT: The capacitances of capacitors are "built in quantities." Thus, C_1 is the same before and after the battery is removed.

 

[scholio]

didn't i already find Q_1 when i did this calculation:
" let C_1 = Q/V_1 where C_1 = 100 pico farads = 100 *10^-12 farads and V_1 = 50 volts
solve for Q to get Q = 5*10^-9 coulombs " ??

also, how do you know that the capacitors are in parallel? if it is parallel, capacitance C = C_1 + C_2 and C_1 = 100 pico farads, how is the change in voltage factored in?

 

[nrqed]

didn't i already find Q_1 when i did this calculation:
" let C_1 = Q/V_1 where C_1 = 100 pico farads = 100 *10^-12 farads and V_1 = 50 volts
solve for Q to get Q = 5*10^-9 coulombs " ??

also, how do you know that the capacitors are in parallel? if it is parallel, capacitance C = C_1 + C_2 and C_1 = 100 pico farads, how is the change in voltage factored in?

Well, this is a situation where, in some sense, the capacitors can be said to be in parallel and they can be said to be in series so it's a bit confusing.

Bu the key point is that they are the only two elements in a loop, right? By Kirchoff's voltage law, the potential difference across each capacitor must be equal (in absolute value) so there is a 35 volts potential difference on each of them

The Q_1 you found (using the 50 volts) gives the total charge on th etwo capacitors after they have been connected. After they are connected Q on capacitor 1 decreases and Q on capacitor two increases so you now have Q_1 + Q_2 = 5*10^-9 coulombs

So you know the sum of the two charges. And you know that for each capacitor you have

Q_1 = C_1 V and Q_2 = C_2 V

(using the fact that they have the same potential difference V).
So you have three equations fo rthree unknowns (Q_1,Q_2 and C_2). So you can solve

 

[scholio]

i am confused about the value of V, is it 35 or 50 volts?
once of found V is this how i would go about finding capacitance C_2:

since Q_1 + Q_2 = 5*10^-9 and Q_1 = C_1V and Q_2 = C_2V :

if i let Q_1 = 5*10^-9 - Q_2
----> sub that in for Q_1 so that 5*10^-9 - Q_2 = C_1V
----> solve that eq for Q_2 so that Q_2 = -C_1V + 5*10^-9
----> sub that into the 'Q_2 = ...' eq and then solve that so that finally

C_2 = (-C_1V + 5*10^-9)/V

did i calculate for C_2 correctly?

 

[G01]

In the part where the battery is connected, it is V is 50 volts. In the second part, if one capacitor has 35 volts on it, the other must have 35 volts on it, as nrqed said.

You equation for C_2 looks good to me.

 

[scholio]

so for solving for C_2 using the equation above, i should use V=35volts, C_1 = 100 picocoulombs? and get:

C_2 = (-C_1V + 5*10^-9)/V = [(100*10^-12)(35) + 5*10^-9]/35 = 2.43*10^-10 farads?

correct now?

 

[alphysicist]

so for solving for C_2 using the equation above, i should use V=35volts, C_1 = 100 picocoulombs? and get:

C_2 = (-C_1V + 5*10^-9)/V = [(100*10^-12)(35) + 5*10^-9]/35 = 2.43*10^-10 farads?

correct now?

A minor point: I think you dropped the minus sign in front of C_1V.

 

[scholio]

okay, other than the sign, my calculation is correct?

thanks for your help