Question

Thee uncharged capacitors of capacitance $${C}_{1},{C}_{2}$$ and $${C}_{3}$$ are connected as shown in figure to one another and to points $$A,B$$ and $$D$$ at potentials $${\phi}_{A},{\phi}_{B}$$ and $${\phi}_{D}$$.
Determine the potential $${\phi}_{0}$$ at point $$O$$.

Answer 1

Taking into account the relation between the capacitance, voltage, and charge of a capacitor, we can write the following equations for the three capacitors:
$$_{ A }-{ \phi }_{ 0 }=\cfrac { { q }_{ 1 } }{ { C }_{ 1 } } ,{ \phi }_{ B }-{ \phi }_{ 0 }=\cfrac { { q }_{ 2 } }{ { C }_{ 2 } } ,{ \phi }_{ D }-{ \phi }_{ 0 }=\cfrac { { q }_{ 3 } }{ { C }_{ 3 } } $$
where $${C}_{1},{C}_{2}$$ and $${C}_{3}$$ are the capacitances of the corresponding capacitors and $${q}_{1},{q}_{2}$$ and $${q}_{3}$$ are the charges on their plates. According to the charge conservation law, $${q}_{1}+{q}_{2}+{q}_{3}=0$$, and hence the potential of the common point $$O$$ is
$${ \phi }_{ 0 }=\cfrac { { \phi }_{ A }{ C }_{ 1 }+{ \phi }_{ B }{ C }_{ 2 }+{ \phi }_{ D }{ C }_{ 3 } }{ { C }_{ 1 }+{ C }_{ 2 }+{ C }_{ 3 } } $$