**Resistors in Series**

Resistors in series
means end to end connection. When they are connected in series, we can find
that the current will be the same across all of them and the supplied voltage
across them is distributed proportional to the resistance.

We can find the
effective resistance of the system when number of resistors are connected in
series and also find out the voltage drop across each resistors as shown below.
The effective resistance increases in series combination.

**Resistors in parallel**

When similar end of
all resistors are connected together and the same with the other ends, the
connection is called parallel connection. In parallel connection, the voltage
across all the elements is same and the current across them is distributed such
that it is inversely proportional to resistance.

We can find the
effective resistance and the current in each element as shown below. The
effective resistance in parallel is less than even the small value of the
circuit. The effective resistance decreases in parallel combination.

**Variation of resistance Problems and Solutions**

We need to find the
effective resistance between the two points as shown in the given picture. We
can simply solve the problem by bisecting the entire circuit into two identical
parts. The two parts are in series with each other and symmetrical. If we are
able to find the resistance of one part, by adding the same value to that, we
can find the total resistance of the system given.

As we have bisected
the resistor into two parts, its length so its resistance also becomes half. By
identifying the resistors in series and parallel and measuring their effective
resistance, we can find the total resistance of the circuit as shown below.

**Problem and Solution**

This problem is
about percentage change in the resistance of a wire when there is a change in
its length alone. No information is given in the problem about its area. As the
volume of the wire remains constant, we need to write area in terms of length
and volume. Thus we can prove that resistance is directly proportional to the
square of the length of the wire.

The problem is
solved as shown below.

**Problem and solution**

This problem is
about variation of resistance with mass of the wire. There is information in
the problem about its length but not about area. We can change the area interms
of mass and solve the problem as shown in the diagram below.

**Problem and solution**

This problem is
about current passing in a wire when multiple wires are connected in parallel.
The total current across the combination is given to us and resistance of each
wire is given. We know that if resistors are connected in parallel, the voltage
across them is same. Hence the current flow is reciprocal to resistance and the
problem can be solved as shown below.

**Problem and solution**

This problem is
about finding a voltage across a resistor when multiple resistors are connected
in the circuit as shown in the diagram.

By identifying the
elements and currents across them, we can solve the problem as shown below.

**Problem and solution**

If two parts of a
circle separated by an angle are two wires having different resistance, we need
to measure the effective resistance of the circuit. The problem is solved as
shown below.

**Problem and solution**

This problem is
about to find the effective resistance of the system where infinite resistors
are connected as shown in the diagram. We can identify the symmetry in the
ladder and we can say that the circuit is the combination of similar
symmetrical parts. Except one part, we can assume that all other parts together
are having some resistance and even with the other remaining ladder, the answer
still remains same. Simply because of infinite ladders, adding or removing one
ladder is not going to make a big difference to the entire system
significantly.

Thus we can solve
the problem as shown below.

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