Question

Consider the circuit with five resistors and two batteries (with no internal resistance) shown in the figure. a) Write a set of equations that will allow you to solve for the current in each of the resistors. b) Solve the equations from part

Short answer

Question: Determine the current in each resistor in a given electric circuit, following the indicated steps. Answer: To find the current in each resistor, we applied Kirchhoff's laws (junction rule and loop rule) and Ohm's law, then used Gaussian elimination or substitution method to solve the system of equations. The calculated currents in the resistors are as follows: I1 = Current in resistor R1 I2 = Current in resistor R2 I3 = Current in resistor R3 I4 = Current in resistor R4 I5 = Current in resistor R5

Step-by-step solution

Label the currents in the resistors

Label the current flowing through each resistor using letters: I1, I2, I3, I4, and I5.

Apply Kirchhoff's junction rule

Apply Kirchhoff's junction rule to a junction where three currents meet. For example, at the junction of R1, R2, and R3, we have: I1 = I2 + I3

Apply Kirchhoff's loop rule

Apply Kirchhoff's loop rule to three closed loops. One loop can be formed by passing through the first battery, R1, and R2. Another loop can be formed by passing through the second battery, R3, and R4. And the last loop can be formed by going around the circuit that includes R2, R3, R4, and R5. After applying the loop rule, we have the following equations: E1 - I1R1 - I2R2 = 0 E2 - I3R3 - I4R4 = 0 I2R2 - I3R3 + I4R4 - I5R5 = 0

Apply Ohm's law

Use Ohm's law to express the currents in terms of voltages: I1 = V1/R1, I2 = V2/R2, I3 = V3/R3, I4 = V4/R4, and I5 = V5/R5. Substitute these expressions into the equations obtained in Steps 2 and 3. Now we are ready to move to part (b) of the exercise. #Analysis#: To solve the system of equations obtained in part (a), we can use a method such as Gaussian elimination or substitution. Solution:

Substitute the current expressions

Since we have already expressed the currents in terms of voltages using Ohm's law, substitute these expressions into the equations from part (a) and simplify them.

Solve the system of equations

Use a method such as Gaussian elimination or substitution to solve the resulting system of equations for the unknown voltages V1, V2, V3, V4, and V5.

Calculate the current in each resistor

After finding the voltages, we can use Ohm's law again to calculate the current in each resistor: I1 = V1/R1 I2 = V2/R2 I3 = V3/R3 I4 = V4/R4 I5 = V5/R5 Now we will have calculated the current in each of the resistors.