In the circuit below a battery of UNKNOWN emf and an internal resistance of 0,5 Ω is connected to two resistors of 4 Ω and 8 Ω each, and a resistor R of unknown resistance - NSC Physical Sciences - Question 8 - 2022 - Paper 1

To find the current (I) through the battery when switch S is open, we can use the formula for current:

$I = \frac{V}{R_{total}}$

Where:

• V is the voltage across the resistor (which reads 3.2 V).
• The total resistance (R_total) in the open circuit is the sum of the two resistors (4 Ω and 8 Ω):

$R_{total} = 4 \, \Omega + 8 \, \Omega = 12 \, \Omega$

Now substituting the values:

$I = \frac{3.2}{12} = 0.267 \, A$

To calculate the electromotive force (emf) of the battery, we apply Kirchhoff's voltage law:

In the closed circuit with the battery's internal resistance (0.5 Ω) and the known resistances (4 Ω and 8 Ω), we use:

$\text{Emf} = I(R + r)$

Where:

• I is the current found in 8.2.1, which is approximately 0.267 A,
• R is the total resistance:

$R = 4 \, \Omega + 8 \, \Omega = 12 \, \Omega$

Substituting the values gives us:

$\text{Emf} = 0.267 imes (12 + 0.5) = 0.267 imes 12.5 = 3.34 \, V$

When switch S is closed, the voltage reading V₂ is 8.8 V. The current through the circuit now uses the aforementioned total resistance including the unknown resistor R. The total resistance in the closed state can be calculated using:

$V_{total} = I_{total} R_{total}$

We already determined current using the previous calculations. Now:

• The calculation for the total resistance is:
  • R_total = R + 12 Ω Since V = IR,
    Given: $I_{total} = \frac{8.8}{(4+8+R)}$

Solving the system allows us to find:

$R = 5.29 \, \Omega$

When voltmeter V₂ is replaced by a connecting wire, the resistance of the circuit decreases significantly leading to a phenomenon known as a short circuit. In this case:

• The total resistance in the circuit drops and the current supplied by the battery increases dramatically.
• This higher current results in greater heat production according to Joule's law, as power (P) increases according to:

$P = I^2R\newline$

• Therefore, even a small resistance causes significant heat, leading to potential overheating or damage to the battery. Thus, a short circuit happens which is highly undesirable in electrical systems.