Problem — true or false on series and parallel

Problem

State whether the following statements are true or false, giving reasons. (a) In a series circuit the current is the same in every component. (b) In a parallel circuit the potential difference is the same across every branch. (c) A larger resistance in parallel with a smaller one reduces the equivalent resistance below that of the smaller one. (d) The EMF of a cell is the potential difference measured across its terminals when it delivers maximum current.

Solution

(a) TRUE. In a series circuit there is a single path for the charge: with no nodes, the current cannot split. By conservation of charge the same $i$ flows through each component one after another.

(b) TRUE. The branches of a parallel combination share the same two end nodes; the potential difference between those two nodes is a single value, so every branch is subject to the same $V$. The currents, on the other hand, split according to $i_k = V/R_k$.

(c) TRUE. For two resistors in parallel $R_\text{eq} = \frac{R_1 R_2}{R_1 + R_2} < R_\text{min}$ Indeed, adding a parallel path offers the current an alternative route: the equivalent resistance is always smaller than the smaller of the two. Example: $R_1 = 2\ \Omega$ in parallel with $R_2 = 100\ \Omega$ gives $R_\text{eq} = 200/102 \approx 1{,}96\ \Omega < 2\ \Omega$.

(d) FALSE. The EMF $\varepsilon$ is the terminal voltage on open circuit, i.e. when the delivered current is zero. Under load, the internal drop $r\,i$ lowers the terminal voltage: $V = \varepsilon - r\,i$. At maximum current (short circuit) the terminal voltage actually tends to zero, not to $\varepsilon$.

$\ev{\text{T, T, T, F}}$

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Topics: Electric circuits Concepts: Resistors in series and parallel