Kvant Physics Problem 25

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Problem

If the potential of the anode of the photoelement is higher than that of the cathode, a current $I=10~\text{А}$ flows through the photoelement (saturation current). Otherwise, no current flows through the photoelement. Neglecting the internal resistances of the batteries, find the voltages across the photoelements in the circuits shown in Figure 1 (the resistance values are given in kilo-ohms).

Fig. 1

Setup and Assumptions

The system consists of a set of DC circuits containing batteries, resistors with given resistances in kilohms, and photoelements. Each photoelement behaves as an ideal asymmetric element with threshold-like behavior. When the potential of its anode exceeds the potential of its cathode, it conducts a constant saturation current $I = 10,\text{A}$. When the anode potential is lower than or equal to the cathode potential, it blocks completely and carries zero current.

All batteries are ideal with zero internal resistance. All resistors are ideal ohmic elements obeying $U = IR$. The circuit is assumed to be in steady state, so all currents and potentials are time-independent. Wires are ideal conductors with zero resistance.

The unknown quantities are the voltages across the photoelements in the circuits shown in Figure 1. The numerical values depend on the exact topology of connections in the figure, which is not provided in the text of the problem statement.

Physical Principles

Ohm’s law for each resistor is written as

$U = IR,$

where $U$ is the voltage drop, $I$ is the current through the resistor, and $R$ is its resistance.

Kirchhoff’s current law is applied at each node, requiring the algebraic sum of currents entering a node to equal zero,

$\sum I = 0.$

Kirchhoff’s voltage law is applied to each closed loop, requiring the algebraic sum of potential differences along any closed contour to vanish,

$\sum U = 0.$

The photoelement is modeled as a nonlinear ideal element with piecewise behavior. In the conducting regime, it imposes a fixed current $I = 10,\text{A}$ from anode to cathode. In the blocking regime, it imposes $I = 0$. The operating regime is determined by the sign of the voltage across its terminals.

Derivation

Let each circuit in Figure 1 be indexed by $k$. For each circuit, the solution procedure begins by assigning node potentials $V_i$ to all junctions relative to a chosen reference node.

Each resistor connecting nodes $i$ and $j$ contributes a current

$I_{ij} = \frac{V_i - V_j}{R_{ij}}.$

Each battery enforces a fixed potential difference between its terminals,

$V_i - V_j = \mathcal{E},$

with polarity determined by the diagram.

Each photoelement introduces a conditional constraint. In the conducting state,

$I_{\text{photo}} = 10,\text{A},$

and the corresponding voltage is an unknown $U_{\text{photo}} = V_a - V_c$ determined from Kirchhoff’s equations. In the blocking state,

$I_{\text{photo}} = 0,$

which reduces the circuit to an open branch.

For each circuit, both operating states of each photoelement must be tested for consistency. The physically valid state is the one satisfying simultaneously the current constraint and the inequality condition on the anode-cathode potential.

After selecting the correct state, Kirchhoff’s current law is applied at all nodes, producing a linear system in the unknown node potentials. Solving this system yields all node voltages, and the photoelement voltage follows as

$U_{\text{photo}} = V_a - V_c.$

Result

A numerical expression for the voltages across the photoelements cannot be obtained without the explicit circuit topology and resistor values shown in Figure 1. The final answer depends on the connectivity of elements and the distribution of resistances and sources in each subcircuit.

For each circuit in Figure 1, the required result is obtained by solving the corresponding Kirchhoff system and evaluating

$U_{\text{photo}} = V_a - V_c,$

with the conducting branch current fixed at $10,\text{A}$ when the anode is at higher potential than the cathode.

Sanity Checks

Dimensional consistency requires that all computed voltages have units of volts, since each term arises from products of amperes and ohms or from battery electromotive forces. Any intermediate expression combining currents and resistances must reduce to volts.

A typical failure mode in this class of problems occurs when the photoelement state is assumed without verifying the resulting sign of $V_a - V_c$. Another common error arises from inconsistent current direction assignments in Kirchhoff’s current law equations, which can invert the inferred operating regime of the nonlinear element.

A further consistency requirement is that in the conducting regime the computed current through the photoelement branch must equal $10,\text{A}$, and in the blocking regime the computed current must vanish exactly. Any deviation indicates an incorrect circuit reduction or algebraic inconsistency in the node equations.