Consider a Wheatstone bridge with resistance and capacitance connected as shown.

Find the condition on the resistance and the capacitance such that the bridge remains balanced at all times.

Solution

Suppose that the bridge is balanced ie., V

AB

=V

AD

;V

BC

=V

DC

Let the current and the charges on the capacitors in the circuit as shown then

i

1

R

1

=i

3

R

3

…(i)

C

2

q

2

=

C

4

q

4

…(ii)

Consider the charging of the part of the circuit shown alongside. Let q

2

be the charge on C

2

and i

1

be the current in the circuit

Then,

i

1

R

1

+

C

2

q

2

=ε…(iii);

dt

dq

2

+

R

2

C

2

q

2

=i

1

…(iv)

Substituting (iv) in (ii) and simplifying

dt

dq

2

+

R

eq

C

2

q

2

=

R

1

ε

[

R

eq

1

=

R

1

1

+

R

2

1

]

∴q

2

=

R

1

εR

eq

C

2

⎣

⎢

⎢

⎢

⎡

1−e

R

eq

C

2

1

⎦

⎥

⎥

⎥

⎤

=

R

1

+R

2

εR

2

C

2

[1−e

−t/R

eq

C

2

]

Similarly for the other circuit, we have

q

4

=

R

3

+R

4

εR

4

C

4

⎣

⎢

⎢

⎢

⎢

⎡

1−e

R

eq

′

C

2

1

⎦

⎥

⎥

⎥

⎥

⎤

[

R

eq

′

1

=

R

3

1

+

R

4

1

]

Now

C

2

q

2

=

C

4

q

4

which leads to

R

1

+R

2

R

2

=

R

3

+R

4

R

4

;

R

eq

C

2

1

=

R

eq

′

C

4

1

or

R

2

R

1

=

R

4

R

3

;R

1

C

2

=R

3

C

4