Given below are two statements:
Statement-I: The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs.
Statement-II: The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries.
In light of the above statements, choose the correct answer from the options given below.
Show Hint
- Statement-I is true but Statement-II is false
- Both Statement-I and Statement-II are false
- Both Statement-I and Statement-II are true
- Statement-I is false but Statement-II is true
The Correct Option is D
Approach Solution - 1
To understand why the correct choice is "Statement-I is false but Statement-II is true", we need to examine each statement individually.
Statement-I: "The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs."
When two nonideal batteries are connected in parallel, the equivalent emf (Eeq) is not simply smaller than either emf but rather an average based on their internal resistances. The formula for the equivalent emf is:
Eeq= (E1R2 + E2R1)/(R1+R2)
Here, E1 and E2 are the emfs of the two batteries, while R1 and R2 are their internal resistances. The Eeq is essentially a weighted average and can be greater than or equal to the smaller emf and smaller than or equal to the larger emf. Therefore, Statement-I is false.
Statement-II: "The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries."
When internal resistances are combined in parallel, the equivalent resistance (Req) is smaller than either of the individual resistances. The formula for the equivalent internal resistance is:
1/Req = 1/R1 + 1/R2
As a result, Req is indeed smaller than either R1 or R2. Therefore, Statement-II is true.
Thus, the correct answer is: Statement-I is false but Statement-II is true.
Approach Solution -2
Statement-I: The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs.
Statement-II: The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries.
Choose the correct answer from the options given below.
Solution
Step 1 — Concept of parallel connection of nonideal batteries:
Let two batteries with emfs $E_1$, $E_2$ and internal resistances $r_1$, $r_2$ be connected in parallel.
The equivalent emf $E_{eq}$ is given by:
$$ E_{eq} = \frac{E_1/r_1 + E_2/r_2}{1/r_1 + 1/r_2} $$
and the equivalent internal resistance $r_{eq}$ is given by:
$$ r_{eq} = \frac{r_1 r_2}{r_1 + r_2} $$
Step 2 — Compare internal resistances:
Since $r_{eq} = \dfrac{r_1 r_2}{r_1 + r_2}$, it is always less than either $r_1$ or $r_2$.
Therefore, Statement-II is true.
Step 3 — Compare emfs:
- If $E_1 = E_2$, then $E_{eq} = E_1 = E_2$.
- If $E_1 \neq E_2$, then $E_{eq}$ is a weighted average of $E_1$ and $E_2$, and hence lies between $E_1$ and $E_2$.
Thus, $E_{eq}$ is not smaller than both; it lies between them.
Hence, Statement-I is false.
Step 4 — Conclusion:
Statement-I is false but Statement-II is true.
Correct Option: 4
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