Final answer:
To calculate the voltage across each resistor and the power consumed, use Ohm's law and power formulas, considering the total voltage supplied and the V-I relationship of the non-linear resistor which leads to a quadratic equation for the current.
Step-by-step explanation:
To find the voltage across each resistor and the power consumed by each, we can follow these steps:
- First, we use Ohm's law (V = IR) and power formulas to understand the relationships between voltage, current, resistance, and power.
- The given V-I relationship for the non-linear resistor is V = 47i², where V is the voltage across the non-linear resistor, and i is the current flowing through it.
- To find the current i, we realize that the total voltage provided by the source (55 V) must equal the sum of the voltages across each resistor when they are in series. Thus, 55 V = V_linear + V_non-linear. Because V_linear = 47i (Ohm's law for the linear resistor), and V_non-linear is given by 47i², we need to solve for i in the equation: 55 = 47i + 47i².
- We can find i by solving the quadratic equation (details of this calculation are omitted for brevity).
- Once we have i, we can calculate the voltage across each resistor using V_linear = 47i and V_non-linear = 47i² (substituting the i obtained from solving the quadratic equation).
- The power consumed by each resistor can be found using the formula P = I²R, where P is the power, I is the current, and R is the resistance.
The voltage across the linear resistor will be V_linear = 47i, and the voltage across the non-linear resistor will be V_non-linear = 47i². Similarly, the power consumed by the linear resistor would be P_linear = i²R_linear, and for the non-linear resistor, P_non-linear = i²R_non-linear (where R_non-linear is a function of current via the given V-I relationship).