Electric Current：練習題

A circuit consists of a battery with electromotive force \(\mathcal{E}\) and internal resistance \(r\) connected to a variable external resistor \(R\). As the resistance \(R\) is increased from a very small value to a very large value, how does the power dissipated by the internal resistance \(r\) and the terminal voltage \(V_{ab}\) across the battery change?

In an RC circuit, a capacitor of capacitance \(C\) is initially charged to a potential difference \(V_0\). At time \(t = 0\), it is connected in series with a resistor of resistance \(R\) and an ideal inductor of inductance \(L\). However, if we ignore the inductor and consider only the RC discharge, what is the ratio of the energy stored in the capacitor at time \(t = RC \ln(2)\) to its initial energy at \(t = 0\)?

In the circuit shown, three identical resistors each of resistance \(R\) and two identical capacitors each of capacitance \(C\) are connected to a battery of emf \(\mathcal{E}\). After the circuit has been connected for a very long time, what is the total energy stored in the two capacitors if they are connected in parallel with one of the resistors?

An ammeter with a resistance of \(0.1\, \Omega\) and a voltmeter with a resistance of \(10\, \text{k}\Omega\) are used to measure a resistance \(R\). In one configuration, the ammeter is in series with the parallel combination of the voltmeter and \(R\). If the voltmeter reads \(100\text{ V}\) and the ammeter reads \(1.01\text{ A}\), what is the actual value of \(R\)?

In an RC circuit, a capacitor \(C\) is being charged by a battery \(\mathcal{E}\) through a resistor \(R\). At the instant the current in the circuit has decreased to \(1/e\) of its initial value, what is the ratio of the energy stored in the capacitor to the total energy supplied by the battery up to that time?

A circuit consists of a DC voltage source \(V_0\), a resistor \(R\), and an inductor \(L\) connected in series. At time \(t = 0\), the switch is closed.
(a) Write the differential equation for the current \(i(t)\) in the circuit.
(b) Solve the equation to find \(i(t)\) as a function of time.
(c) Determine the time at which the energy stored in the inductor is half of its maximum steady-state value.

Consider a network of five identical resistors each of resistance \(R\) connected in a Wheatstone bridge configuration. A voltage source \(V\) is applied across the input terminals.
(a) If the bridge is unbalanced because one resistor is changed to \(2R\), calculate the equivalent resistance of the entire network.
(b) Determine the current flowing through each branch using Kirchhoff's rules.