# 21.3 Kirchhoff’s rules  (Page 3/8)

 Page 3 / 8

## Calculating current: using kirchhoff’s rules

Find the currents flowing in the circuit in [link] .

Strategy

This circuit is sufficiently complex that the currents cannot be found using Ohm’s law and the series-parallel techniques—it is necessary to use Kirchhoff’s rules. Currents have been labeled ${I}_{1}$ , ${I}_{2}$ , and ${I}_{3}$ in the figure and assumptions have been made about their directions. Locations on the diagram have been labeled with letters a through h. In the solution we will apply the junction and loop rules, seeking three independent equations to allow us to solve for the three unknown currents.

Solution

We begin by applying Kirchhoff’s first or junction rule at point a. This gives

${I}_{1}={I}_{2}+{I}_{3},$

since ${I}_{1}$ flows into the junction, while ${I}_{2}$ and ${I}_{3}$ flow out. Applying the junction rule at e produces exactly the same equation, so that no new information is obtained. This is a single equation with three unknowns—three independent equations are needed, and so the loop rule must be applied.

Now we consider the loop abcdea. Going from a to b, we traverse ${R}_{2}$ in the same (assumed) direction of the current ${I}_{2}$ , and so the change in potential is $-{I}_{2}{R}_{2}$ . Then going from b to c, we go from $–$ to +, so that the change in potential is $+{\text{emf}}_{1}$ . Traversing the internal resistance ${r}_{1}$ from c to d gives $-{I}_{2}{r}_{1}$ . Completing the loop by going from d to a again traverses a resistor in the same direction as its current, giving a change in potential of $-{I}_{1}{R}_{1}$ .

The loop rule states that the changes in potential sum to zero. Thus,

$-{I}_{2}{R}_{2}+{\text{emf}}_{1}-{I}_{2}{r}_{1}-{I}_{1}{R}_{1}=-{I}_{2}\left({R}_{2}+{r}_{1}\right)+{\text{emf}}_{1}-{I}_{1}{R}_{1}=0.$

Substituting values from the circuit diagram for the resistances and emf, and canceling the ampere unit gives

$-{3I}_{2}+\text{18}-{6I}_{1}=0.$

Now applying the loop rule to aefgha (we could have chosen abcdefgha as well) similarly gives

$+\phantom{\rule{0.25em}{0ex}}{I}_{1}{R}_{1}+{I}_{3}{R}_{3}+{I}_{3}{r}_{2}-{\text{emf}}_{2}\text{= +}{I}_{1}{R}_{1}+{I}_{3}\left({R}_{3}+{r}_{2}\right)-{\text{emf}}_{2}=0.$

Note that the signs are reversed compared with the other loop, because elements are traversed in the opposite direction. With values entered, this becomes

$+\phantom{\rule{0.25em}{0ex}}{6I}_{1}+{2I}_{3}-\text{45}=0.$

These three equations are sufficient to solve for the three unknown currents. First, solve the second equation for ${I}_{2}$ :

${I}_{2}=6-{2I}_{1}.$

Now solve the third equation for ${I}_{3}$ :

${I}_{3}=\text{22}\text{.}5-{3I}_{1}.$

Substituting these two new equations into the first one allows us to find a value for ${I}_{1}$ :

${I}_{1}={I}_{2}+{I}_{3}=\left(6-{2I}_{1}\right)+\left(\text{22}\text{.}5-{3I}_{1}\right)=\text{28}\text{.}5-{5I}_{1}.$

Combining terms gives

${6I}_{1}=\text{28}\text{.}5, and$
${I}_{1}=4\text{.}\text{75 A}.$

Substituting this value for ${I}_{1}$ back into the fourth equation gives

${I}_{2}=6-{2I}_{1}=6-9.50$
${I}_{2}=-3\text{.}\text{50 A}.$

The minus sign means ${I}_{2}$ flows in the direction opposite to that assumed in [link] .

Finally, substituting the value for ${I}_{1}$ into the fifth equation gives

${I}_{3}=\text{22.5}-{3I}_{1}=\text{22.5}-\text{14}\text{.}\text{25}$
${I}_{3}=8\text{.}\text{25 A}.$

Discussion

Just as a check, we note that indeed ${I}_{1}={I}_{2}+{I}_{3}$ . The results could also have been checked by entering all of the values into the equation for the abcdefgha loop.

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
What is thermal heat all about
why uniform circular motion is called a periodic motion?.
when a train start from A & it returns at same station A . what is its acceleration?
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
what are the types of radioactivity
Worku
what is static friction
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
definition of mass of conversion
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
Boniface
the range of objects and phenomena studied in physics is
Boniface