A) ![C_3 < C_1 < C_2](https://tex.z-dn.net/?f=C_3%20%3C%20C_1%20%3C%20C_2)
The capacitance of a parallel-plate capacitor is given by
![C=\epsilon_0 \frac{A}{d}](https://tex.z-dn.net/?f=C%3D%5Cepsilon_0%20%5Cfrac%7BA%7D%7Bd%7D)
where
A is the plate area
d is the plate separation
Here we have:
- Capacitor 1: plate area A, plate separation d
capacitance: ![C_1=\epsilon_0 \frac{A}{d}](https://tex.z-dn.net/?f=C_1%3D%5Cepsilon_0%20%5Cfrac%7BA%7D%7Bd%7D)
- Capacitor 2: plate area 2A, plate separation d
capacitance: ![C_2=\epsilon_0 \frac{2A}{d} = 2C_1](https://tex.z-dn.net/?f=C_2%3D%5Cepsilon_0%20%5Cfrac%7B2A%7D%7Bd%7D%20%3D%202C_1)
- Capacitor 3: plate area A, plate separation 2d
capacitance: ![C_3=\epsilon_0 \frac{A}{2d}=\frac{C_1}{2}](https://tex.z-dn.net/?f=C_3%3D%5Cepsilon_0%20%5Cfrac%7BA%7D%7B2d%7D%3D%5Cfrac%7BC_1%7D%7B2%7D)
So ranking the three capacitor from least to greatest capacitance we have:
![C_3 < C_1 < C_2](https://tex.z-dn.net/?f=C_3%20%3C%20C_1%20%3C%20C_2)
2. ![V_2 < V_1 < V_3](https://tex.z-dn.net/?f=V_2%20%3C%20V_1%20%3C%20V_3)
The three capacitors have same amont of charge, Q.
The potential difference between the plates on each capacitor is given by
![V = \frac{Q}{C}](https://tex.z-dn.net/?f=%20V%20%3D%20%5Cfrac%7BQ%7D%7BC%7D)
so here we have
- Capacitor 1: ![C = C_1](https://tex.z-dn.net/?f=C%20%3D%20C_1)
Potential difference: ![V_1 = \frac{Q}{C_1}](https://tex.z-dn.net/?f=%20V_1%20%3D%20%5Cfrac%7BQ%7D%7BC_1%7D)
- Capacitor 2: ![C = 2C_1](https://tex.z-dn.net/?f=C%20%3D%202C_1)
Potential difference: ![V_2= \frac{Q}{2C_1}=\frac{ V_1}{2}](https://tex.z-dn.net/?f=%20V_2%3D%20%5Cfrac%7BQ%7D%7B2C_1%7D%3D%5Cfrac%7B%20V_1%7D%7B2%7D)
- Capacitor 3: ![C = \frac{C_1}{2}](https://tex.z-dn.net/?f=C%20%3D%20%5Cfrac%7BC_1%7D%7B2%7D)
Potential difference: ![V_3 = \frac{Q}{C_1/2}=2 V_1](https://tex.z-dn.net/?f=%20V_3%20%3D%20%5Cfrac%7BQ%7D%7BC_1%2F2%7D%3D2%20V_1%20)
So ranking the three capacitor from least to greatest potential difference we have:
![V_2 < V_1 < V_3](https://tex.z-dn.net/?f=V_2%20%3C%20V_1%20%3C%20V_3)
C. ![E_2 < E_1 = E_3](https://tex.z-dn.net/?f=E_2%20%3C%20E_1%20%3D%20E_3)
The electric field magnitude between the plates of a capacitor is given by
![E=\frac{V}{d}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7BV%7D%7Bd%7D)
where V is the potential difference between the plates and d is the plate separation
So here we have
- Capacitor 1: potential difference
, plate separation d
electric field: ![E_1 = \frac{V_1}{d}](https://tex.z-dn.net/?f=E_1%20%3D%20%5Cfrac%7BV_1%7D%7Bd%7D)
- Capacitor 2: potential difference
, plate separation d
electric field: ![E_2=\frac{V_1/2}{d} =\frac{V_1}{2d}= \frac{E_1}{2}](https://tex.z-dn.net/?f=E_2%3D%5Cfrac%7BV_1%2F2%7D%7Bd%7D%20%3D%5Cfrac%7BV_1%7D%7B2d%7D%3D%20%5Cfrac%7BE_1%7D%7B2%7D)
- Capacitor 3: potential difference
, plate separation 2d
electric field: ![E_3=\frac{2 V_1}{2d} =\frac{V_1}{d}= E_1](https://tex.z-dn.net/?f=E_3%3D%5Cfrac%7B2%20V_1%7D%7B2d%7D%20%3D%5Cfrac%7BV_1%7D%7Bd%7D%3D%20E_1)
So ranking the three capacitor from least to greatest electric field we have:
![E_2 < E_1 = E_3](https://tex.z-dn.net/?f=E_2%20%3C%20E_1%20%3D%20E_3)
D. ![U_2 < U_1 < U_3](https://tex.z-dn.net/?f=U_2%20%3C%20U_1%20%3C%20U_3)
The energy stored in a capacitor is
![U=\frac{1}{2}QV](https://tex.z-dn.net/?f=U%3D%5Cfrac%7B1%7D%7B2%7DQV)
where Q is the same for the three capacitors
Here we have
- Capacitor 1: potential difference ![V_1](https://tex.z-dn.net/?f=V_1)
energy: ![U_1 = \frac{1}{2}QV_1](https://tex.z-dn.net/?f=U_1%20%3D%20%5Cfrac%7B1%7D%7B2%7DQV_1)
- Capacitor 2: potential difference ![\frac{V_1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7BV_1%7D%7B2%7D)
energy: ![U_2 = \frac{1}{2}Q\frac{V_1}{2}=\frac{U_1}{2}](https://tex.z-dn.net/?f=U_2%20%3D%20%5Cfrac%7B1%7D%7B2%7DQ%5Cfrac%7BV_1%7D%7B2%7D%3D%5Cfrac%7BU_1%7D%7B2%7D)
- Capacitor 3: potential difference ![2V_1](https://tex.z-dn.net/?f=2V_1)
energy: ![U_3 = \frac{1}{2}Q(2 V_1)=2 U_1](https://tex.z-dn.net/?f=U_3%20%3D%20%5Cfrac%7B1%7D%7B2%7DQ%282%20V_1%29%3D2%20U_1)
So ranking the three capacitor from least to greatest energy we have:
![U_2 < U_1 < U_3](https://tex.z-dn.net/?f=U_2%20%3C%20U_1%20%3C%20U_3)
E. ![u_2 < u_1 = u_3](https://tex.z-dn.net/?f=u_2%20%3C%20u_1%20%3D%20u_3)
The energy density in a capacitor is given by
![u=\frac{1}{2}\epsilon_0 E^2](https://tex.z-dn.net/?f=u%3D%5Cfrac%7B1%7D%7B2%7D%5Cepsilon_0%20E%5E2)
where E is the electric field strength
Here we have
- Capacitor 1: electric field ![E_1](https://tex.z-dn.net/?f=E_1)
Energy density: ![u_1=\frac{1}{2}\epsilon_0 E_1^2](https://tex.z-dn.net/?f=u_1%3D%5Cfrac%7B1%7D%7B2%7D%5Cepsilon_0%20E_1%5E2)
- Capacitor 2: electric field ![\frac{E_1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7BE_1%7D%7B2%7D)
energy density: ![u_2=\frac{1}{2}\epsilon_0 (\frac{E_1}{2})^2=\frac{E_1}{4}](https://tex.z-dn.net/?f=u_2%3D%5Cfrac%7B1%7D%7B2%7D%5Cepsilon_0%20%28%5Cfrac%7BE_1%7D%7B2%7D%29%5E2%3D%5Cfrac%7BE_1%7D%7B4%7D)
- Capacitor 3: electric field ![E_1](https://tex.z-dn.net/?f=E_1)
Energy density: ![u_3=\frac{1}{2}\epsilon_0 E_1^2](https://tex.z-dn.net/?f=u_3%3D%5Cfrac%7B1%7D%7B2%7D%5Cepsilon_0%20E_1%5E2)
So ranking the three capacitor from least to greatest energy density we have:
![u_2 < u_1 = u_3](https://tex.z-dn.net/?f=u_2%20%3C%20u_1%20%3D%20u_3)