_{1}and the current flowing in lower loop is I

_{2}. Let the direction of flow is same (in clockwise direction) for both currents as shown in fig.2.4. Then applying KVL in upper loop, we get:

-10 -2I

_{1}-1(I_{1}–I_{2}) +12 = 0
Or 3I

_{1}–I_{2}= 2 ….. (1)
Similarly, applying KVL in lower loop, we get:

-12 -1(I

_{2}–I_{1}) -5 I_{2}= 0
Or I

_{1}-6I_{2}= 12 …. (2)
Solving these equations, we get

I

_{1}= 0 and I_{2}= 2 A.
In this way in Maxwell’s Loop current method, loop current is calculated to find all the branch currents.

*Some helpful points:*- Assume that all the loop currents are flowing in same direction (say in clockwise).
- When writing loop equation (KVL) on N
^{th}loop, assume the current flowing in that loop i.e. I_{n}is largest current. See on above example: For loop (1) ““ and for loop (2) “*-1(I*_{1}–I_{2})“ is taken because I*-1(I*_{2}–I_{1})_{1 }is assumed as largest current in loop (1) and I_{2}I_{1 }is assumed as largest current in loop (2).

*#Case:*

*If a current source is present in the perimeter of the loop:*
If there is a current source in the perimeter of the N

^{th}loop, then the current flowing through that loop (I_{n}) is equal to the magnitude of the current of the current source.
In this case, as shown in fig.2.5, there is a current source on the perimeter of loop (1). Hence I

_{1}equals**because the direction of I***-2A*_{1}(assumed) is opposite to the direction of source current.

*#Case:*

*If a current source is present on the common branch of two loops:*
If a current source is present on the common branch of M

^{th}and N^{th}loop, then sum or difference of currents flowing on those loops is equal to the magnitude of the current source present. And a new loop is considered to obtain the equation of loop which is known as super-loop. Super-loop is made by mixing those loops which has common current source (i.e. M_{th}and N^{th}loop) without considering that common branch.
For loop (1) and (3)

And from super loop:

By solving these three equations we can get the unknown currents.

###
Cramer’s rule and matrix model

Cramer’s rule is used for the solution of linear equations by determinants. Let the system of linear equations in three unknown I

_{1},I_{2}and I_{3}are:
May be written in matrix from as:

Where R

_{ii}= the total resistance of the i^{th}loop with +ve sign.
R

_{ij}= common resistance between i^{th}loop and j^{th}loop with –ve sign.
Then by Cramer’s rule, the solution of these simultaneous equations is given by:

Where,

Cramer’s rule can be used for solving simultaneous equations if the numbers of unknown are more than two. This gives quick results.