Download kirchoff

Physics - Casao
Montwood High School
Kirchhoff's Rules
Kirchhoff's Rules:
1. The point rule is an application of the principle of conservation of electric charge. It states that the
algebraic sum of the currents toward any branch point is zero, ΣI  0 . Consider those currents
entering a point as positive; those leaving as negative.
2.
By convention, the direction associated with the current in a circuit is the direction in which a positive
charge would move in the potential difference provided by the battery. Thus, the current in the
external circuit is said to be directed from the positive terminal to the negative terminal of the battery.
In solids, such as wires, the mobile charges that are actually free to flow, and therefore make up the
current, are the negatively charged electrons. The motion of the electrons is opposite the conventional
direction assigned to the current. However, since a flow of positive charges in one direction is
equivalent to a flow of negative charges in the opposite direction, we use the convention of positive
current in almost all cases.
3.
In many instances, the current that moves through a branch is the resultant of current produced by
two electron sources. If the direction of the two individual currents is the same, the resultant current
is the direction of the currents produced by the electron sources. If the direction of the two individual
currents is not the same, the resultant current is in the direction of the current produced by the
electron source with the greater potential difference.
4.
The loop rule is an expression of conservation of energy. It states that the algebraic sum of the
potential differences in any loop must equal zero, ΣE  I  R   0 . Thus the potential differences
associated with EMF's and those of resistive elements must be included. The procedure to follow in
using the loop rule can be broken down into the following steps:
 Label all quantities, both known and unknown. Include an assumed sense of direction for
each unknown current and EMF. If a particular direction is chosen incorrectly, the value of
the quantity will emerge from the analysis with a negative sign.
5.

Choose any closed loop in the network and designate a direction, clockwise or
counterclockwise, to traverse the loop.

Go around the loop in the designated direction, adding EMF's and IR terms. An EMF is
considered positive when it is traversed in the same direction as the EMF arrow (from - to +)
and negative when it is traversed in the opposite direction. An IR term is considered
negative if the resistor is traversed in the same direction as the assumed current; positive if in
the opposite direction.

Use ΣE  I  R   0 ; equate the sum of EMF and IR terms from the previous step to zero.

If necessary, choose another loop to obtain a different relationship between the unknowns
and continue until there are as many equations as unknowns or until every circuit element
has been included in at least one of the loops.
Alternative Method (for circuits that are not already labeled):
 For a loop containing one source of EMF, label the positive terminal and the negative
terminal. Go around the loop from the positive terminal and when you arrive at a resistor R,
label this end as positive and the other end as negative. Continue around the loop and do
this for all resistors encountered in the loop until you arrive at the negative terminal of the
source of EMF.
Physics - Casao




2
Kirchhoff’s Rules
If two sources of EMF are present in the loop, the direction of the current and the positive and
negative ends of the resistors is determined by the largest source of EMF.
o Label the positive and negative terminals of all the sources of EMF.
o From the positive terminal of the largest source of EMF, go around the loop from the
positive terminal and when you arrive at a resistor R, label this end as positive and
the other end of the resistor as negative. Do this for all resistors encountered in the
loop until you arrive at the negative terminal of the largest source of EMF.
For a resistor located on a branch that is common to two loops, each with a source of EMF,
travel the branch in the direction of the current produced by the largest source of EMF and
when you arrive at a resistor R, label this end as positive and the other end as negative. Do
this for all the resistors on the shared branch.
For a circuit with two or more loops, but only one source of EMF, travel all loops in the
direction of the current produced by the source of EMF (from the positive terminal to the
negative terminal).
In traveling the loop to write the equation for ΣE  I  R   0 :
o When going across a source of EMF from positive to negative, it is a decrease in
potential and is written as –E in the equation.
o When going across a source of EMF from positive to negative, it is an increase in
potential and is written as +E in the equation.
o When traveling across a resistor R from positive to negative, it is a decrease in
potential and is written as -IR in the equation.
o When traveling across a resistor R from negative to positive, it is an increase in
potential and is written as +IR in the equation.
Physics - Casao
6.
7.
3
Kirchhoff’s Rules
For batteries with an internal resistance r, the EMF across the battery terminals and the voltage drop
(IR) due to the internal resistance must have opposite signs.
 If the source of EMF is traveled from negative to positive (- to +), the EMF term is positive
(+EMF) and the IR term is negative (-IR).
 If the source of EMF is traveled from positive to negative (+ to -), the EMF term is negative
(-EMF) and the IR term is positive (+IR).
There are limitations to the number of times you can use the junction rule and the loop rule.
 The junction rule can be used as often as needed so long as each time you write an equation,
you include in it a current that has not been used in a previous junction rule equation. In
general, the number of times the junction rule must be used is one fewer than the number of
junction points in the circuit.
 The loop rule can be used as often as needed so long as a new circuit element (resistor or
battery) or a new current appears in each new equation. In general, the number of
independent equations you need equals the number of unknowns in order to solve a
particular circuit problem.
Answer these problems on your own paper, preferably notebook paper. All problem solutions
are to include the work for every calculation necessary to resolve the problem; i.e. no “miracles”. Full
credit for the problem will include credit for the intermediate steps; therefore, all intermediate steps must
be present in order to receive full credit.
Problems:
1. Figure 1 shows a circuit that contains two resistors in parallel and a third resistor in series with the
battery. The current entering junction point P is 2 A. The current I1 leaving the junction is -1.2 A.
Determine:
a. the current I2, and
b. the unknown resistor R2.
2.
The battery in Figure 1 has an unknown internal resistance. Determine:
a. the external voltage of the circuit, and
b. the internal resistance of the battery.
3.
Figure 2 shows a junction point in a circuit. The current I1 entering junction point A is 5 A. Two of
the three currents leaving the junction are shown. Find the unknown current I4.
4.
Figure 3 shows a closed circuit that contains two seats of EMF and two resistors. Find the voltage
drop across resistor R1.
5.
Figure 4 shows a closed circuit that contains two sources of EMF with their internal resistance and
resistances in the circuit. Determine the potentials at points a through g in the circuit assuming that
Physics - Casao
4
Kirchhoff’s Rules
the potential at point f is 0 V. Use the following values: E1 = 9 V; E2 = 6 V; r1 = 0.5 ; r2 = 0.5 ;
= 2 ; R2 = 3 ; and R3 = 2.
R1
6.
Determine the current in the circuit illustrated in Figure 5 by applying Kirchhoff's loop rule in the
clockwise direction. What does the negative sign indicate?
7.
Determine the three currents indicated in the circuit illustrated in Figure 6. Neglect the internal
resistances of the batteries.
8.
Determine the three currents indicated in the circuit illustrated in Figure 7. Neglect the internal
resistances of the batteries. Is point A or point B at the higher potential?
9.
Determine E1, E2, and Vab as indicated in the circuit illustrated in Figure 8. Which point, a or b, is at
the higher electric potential? Is point a or point b at the higher potential?
10. Determine the current in each of the branches of the circuit shown in Figure 9. Be sure to clearly
designate the current directions to be used within the equations.
11. Determine the current in each of the branches of the circuit shown in Figure 10. Be sure to clearly
designate the current directions to be used within the equations.
12. A dead battery is “charged” by connecting it to a live battery of another car, as shown in Figure 11.
Determine the current in the starter, the dead battery, and the live battery. Be sure to clearly
designate the current directions to be used within the equations.
13. Determine the current through each resistor in Figure 12.