Finding Currents Using Kirchhoff's Laws

In summary: I1=5AI2=3AI3=1AIn summary, Kirchhoff's Laws were used to solve for the current in the three batteries.
  • #1
Angie K.
108
1

Homework Statement


P26_34.jpg
a. Determine the currents I1, I2, and I3 in the above figure. Assume the intermal resistance of each battery is r = 2.48Ω.

Homework Equations


Kirchhoff's Laws (Junction Rule and Loop Rule)

The Attempt at a Solution


Using Kirchhoff's Laws,
I ended up with three equations:
i1-i2-i3=0
-12i1-2.48i2=4.48
2.48i2-2.48i3=29

Using matrices to find i1, i2 and i3 I come up with answers but they are wrong. I am wondering if there is a mistake that I made and am not seeing it?
 
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  • #2
Your KCL equation looks okay, but your KVL equations don't look right. Can you give more detail about how you arrived at them? Perhaps write them out showing all contributing terms as you do your "KVL walk" around the loop.
 
  • #3
Angie K. said:
i1-i2-i3=0
This looks good.

Angie K. said:
-12i1-2.48i2=4.48
2.48i2-2.48i3=29
How did you get these two equations?
 
  • #4
The Loop Rule Equations:

(Starting from the top, to the left of the 12V battery and going counterclockwise) -2.48-8+12-2.48i2-10-12i+12=0

(Starting from the junction of the right) 10+2.48i2-12-15+6-2.48i3-18=0
 
  • #5
Angie K. said:
(Starting from the top, to the left of the 12V battery and going counterclockwise) -2.48-8+12-2.48i2-10-12i+12=0
For one thing, you forgot to multiply the internal resistance by the current to get the voltage drop.
 
  • #6
Angie K. said:
The Loop Rule Equations:

(Starting from the top, to the left of the 12V battery and going counterclockwise) -2.48-8+12-2.48i2-10-12i+12=0

(Starting from the junction of the right) 10+2.48i2-12-15+6-2.48i3-18=0
I'm seeing a mix of voltage and resistance terms, some resistances multiplied by currents and some not. So as written those equations cannot be correct. You are summing potential changes around the loop, so any resistor value must be multiplied by a current in order to realize its potential change (Ohm's Law). Perhaps you could group the resistance terms for given currents in parentheses?
 
  • #7
Angie K. said:
The Loop Rule Equations:

(Starting from the top, to the left of the 12V battery and going counterclockwise) -2.48-8+12-(2.48i2)-10-(12i1)+12=0

(Starting from the junction of the right) 10+(2.48i2)-12-15+6-(2.48i3)-18=0
 
  • #8
(Starting from the top, to the left of the 12V battery and going counterclockwise) -2.48-8+12-(2.48i2)-10-(12i1)+12=0
Aren't 2.48 and 8 and 10 all resistances? What currents multiply them? I see only two voltage sources in the loop (both 12 V).

(Starting from the junction of the right) 10+(2.48i2)-12-15+6-(2.48i3)-18=0
Again, I don't see currents multiplying each resistance.
 
  • #9
Angie K. said:
The Loop Rule Equations:

(Starting from the top, to the left of the 12V battery and going counterclockwise) -2.48i1-8i1+12-2.48i2-10i2-12i+12=0

(Starting from the junction of the right) 10i2+2.48i2-12-15i3+6-2.48i3-18i3=0
 
  • #10
That looks better! What do you find for your currents?
 
  • #11
Got it, thanks! I just needed to be consistent with labeling which current goes with which resistor.
 

1. What is Kirchhoff's Law?

Kirchhoff's Laws are two principles used in circuit analysis to determine the currents and voltages in a circuit. The first law, also known as Kirchhoff's Current Law (KCL), states that the sum of all currents entering a node in a circuit is equal to the sum of all currents leaving that node. The second law, also known as Kirchhoff's Voltage Law (KVL), states that the sum of all voltages around a closed loop in a circuit is equal to zero.

2. How do Kirchhoff's Laws help us find currents?

Kirchhoff's Laws provide a systematic approach to analyzing a circuit and determining the currents flowing through it. By applying KCL and KVL to different parts of a circuit, we can create a system of equations that can be solved to find the currents in each branch of the circuit.

3. What is the difference between series and parallel circuits?

In a series circuit, all components are connected in a single loop, so the same current flows through each component. In a parallel circuit, the components are connected in multiple branches, so the total current is divided among the branches. Kirchhoff's Laws can be applied to both types of circuits, but the equations used may differ.

4. Can Kirchhoff's Laws be used for non-linear circuits?

Kirchhoff's Laws are based on the principles of conservation of charge and energy, so they can be applied to any circuit regardless of its linearity or complexity. However, the equations used to solve non-linear circuits may be more complex and require advanced mathematical techniques.

5. Are there any limitations to using Kirchhoff's Laws?

Kirchhoff's Laws assume ideal conditions, such as constant voltage sources and ideal circuit components. In real-world circuits, these conditions may not hold true, which can lead to discrepancies between the calculated and actual currents. Additionally, Kirchhoff's Laws only apply to DC circuits, and different principles must be used for AC circuits.

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