Question 89: Current Electricity

Question

Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers Rc < R/2, which of the following statement(s) about anyone of the galvanometers is(are) true?

Accepted Answer

Let i
g
be the current that gives full deflection of the galvanometer. When all components are connected in series (see figure), effective resistance of the circuit is $${R_e} = 2R + 2{R_c}$$ and maximum current allowed in the circuit is i
g
.
Thus, the voltage between A and B is
$${V_{AB}} = {i_g}{R_e} = 2{i_g}(R + {R_c})$$.
Consider the case when two resistors and one galvanometer are connected in series and the second galvanometer is connected in parallel.
The maximum current through each galvanometer is i
g
and the maximum current through the resistors is 2i
g
. Apply Kirchhoff's law to get the voltage between A and B as
$$V{'_{AB}} = 2{i_g}R + 2{i_g}R + {i_g}{R_c}$$
$$ = 2{i_g}(R + {R_c}) + 2{i_g}(R - {R_c}/2)$$
$$ = {V_{AB}} + 2{i_g}(R - {R_c}/2)$$
$$ > {V_{AB}}$$ ($$\because$$ $${R_c} < R/2$$).
Consider the case when all four components are connected in parallel.
Let i and i
g
be the currents through the resistors and the galvanometers. By Kirchhoff's law, $$iR = {i_g}{R_c}$$, which gives $$i = {i_g}{R_c}/R$$. The current between A and B is
$${I_{AB}} = 2i + 2{i_g} = 2{i_g}(1 + {R_c}/R)$$.
Consider the case when the two galvanometers are connected in series and the combination is connected in parallel with both the resistors.
$$I{'_{AB}} = {i_g} + 2i = {i_g} + 2{i_g}(2{R_c}/R)$$
$$ = 2{i_g}(1 + {R_c}/R) - (1 - 2{R_c}/R){i_g}$$
$$ = {I_{AB}} - (1 - 2{R_c}/R){i_g}$$
$$ < {I_{AB}}$$ ($$\because$$ $${R_c} < R/2$$).

Answer

The maximum voltage range is obtained when all the components are connected in series

Answer

The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer

Answer

The maximum current range is obtained when all the components are connected in parallel

Answer

The maximum current range is obtained when the two galvanometers are connected in series, and the combination is connected in parallel with both the resistors

Current Electricity

Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers Rc < R/2, which of the following statement(s) about anyone of the galvanometers is(are) true?

Choose the correct answer: