In a circuit with three series-connected resistances, the resistance of the resistors is r1 = 4MΩ, r2 = 8MΩ, r3 = 12MΩ
In a circuit with three series-connected resistances, the resistance of the resistors is r1 = 4MΩ, r2 = 8MΩ, r3 = 12MΩ, the voltage across the second resistor is U2 = 24V. Determine I, U, R total. Make up the power balance of the circuit.
For a circuit with three series-connected resistances, the following equalities are fulfilled: I total. = I₁ = I₂ = I₃ – for current strength; U total = U₁ + U₂ + U₃ – for voltage; R total = r₁ + r₂ + r₃ – for resistance.
From the condition of the problem it is known that the resistance of the resistors r₁ = 4 MΩ = 4 ∙ 10 ^ 6 Ohm, r₂ = 8 MΩ = 8 ∙ 10 ^ 6 Ohm, r₃ = 12 MΩ = 12 ∙ 10 ^ 6 Ohm, the voltage across the second resistor is U₂ = 24 V. Then the resistance of the circuit will be:
R total = 4 ∙ 10 ^ 6 Ohm + 8 ∙ 10 ^ 6 Ohm + 12 ∙ 10 ^ 6 Ohm; R total = 24 ∙ 10 ^ 6 Ohm.
The current in the circuit is determined by Ohm’s law I₂ = U₂ / r₂, we get:
I₂ = 24 V / 8 ∙ 10 ^ 6 Ohm; I₂ = 3 ∙ 10 ^ (- 6) A, then I total. = I₁ = I₃ = 3 ∙ 10 ^ (- 6) A. The circuit voltage is found by the formula U total. = I total ∙ R total or
U total = 3 ∙ 10 ^ (- 6) A ∙ 24 ∙ 10 ^ 6 Ohm; U total = 72 V.
The voltage across the first resistor will be: U₁ = I₁ ∙ r₁; U₁ = 3 ∙ 10 ^ (- 6) A ∙ 4 ∙ 10 ^ 6 Ohm = 12 V. At the third resistor, the voltage will be: U₃ = U total. – (U₁ + U₂); U₃ = 72 V – (12 V + 24 V); U₃ = 36 V.
Let’s compose the power balance of the circuit, that is, check the equality:
R total. = Р₁ + Р₂ + Р₃.
To do this, we find P total. = I total. ∙ U total; P = 3 ∙ 10 ^ (- 6) A ∙ 72 V = 216 μW.
For resistors:
Р₁ = I₁ ∙ U₁; Р₁ = 3 ∙ 10 ^ (- 6) А ∙ 12 V; Р₁ = 36 μW;
Р₂ = I₂ ∙ U₂; Р₂ = 3 ∙ 10 ^ (- 6) А ∙ 24 V; Р₂ = 72 μW;
Р₃ = I₃ ∙ U₃; Р₃ = 3 ∙ 10 ^ (- 6) A ∙ 36 V; Р₁ = 108 μW;
216 μW = 36 μW + 72 μW + 108 μW – true equality, which means, indeed, the sum of the powers consumed by the receivers is equal to the power given by the source.
Answer: the total resistance of the circuit is 24 MΩ, the current in the circuit is 3 μA, the voltage of the circuit is 72 V.
