\(P=\cfrac { { V }^{ 2 } }{ R } \Rightarrow R=\cfrac { { V }^{ 2 } }{ P } R\alpha \cfrac { 1 }{ P } \)
P - power, R - resistance
25 W bulb has more resistance. In the series circuit same circuit flows through both the bulbs. The 25 W bulb develops more heat (H = I² Rt) and hence burns out almost instantaneously. 

In series the current, I is same through both wires Rate of production of heat \(P={ I }^{ 2 }R\Rightarrow P\alpha R\) or \(H={ I }^{ 2 }Rt\Rightarrow H\alpha R\)
H - amount of heat produced; R - resistance of the wire.
Nichrome wire is heated up more since resistivity is more and hence resistance of nichrome wire is much more than that of copper wire of same length & radius.

When resistors are connected in series, there is an increase in the effective length. Since resistance varies directly proportional to the length the effective resistance increases \(R\ \alpha\ l\) .

Resistivity = \(\cfrac { m }{ { ne }^{ 2 }\tau } =\rho \)
conductivity = \(\cfrac { { ne }^{ 2 }\tau }{ m } =\cfrac { I }{ \rho } \)
with increase in temperature r decreases. The conductivity decreases 

The drift velocity is defined as the average velocity with which free electrons in a conductor get drifted under the influence of an external electricity applied across the conductor. The relation between drift velocity of free electrons the circuit flowing through the conductor is \(I=\eta Ae{ v }_{ d }e-\) charge of the electron n - no of free electrons in the conductor; A - area of cross-section. 

Furnaces used to manufacture a large number of technologically important materials such as steel, silicon carbide, quartz, gallium arsenide, etc. To produce temperatures up to 1500°C, a molybdenum-nichrome wire wound on a silica tube is used. Carbon arc furnaces produce temperatures up to 3000^oC.

The only disadvantage with the above fuses is that once fuse wire is burnt due to excessive current, they need to be replaced circuit breakers (trippers) are used instead of fuses Whenever there is an excessive current produced due to a faulty wire connection, the circuit breaker switch opens. After repairing the faulty connection, we can close the circuit breaker switch is closed.

The resistanct is defined as the ratio of potential difference across the given conductor to the current passing through the conductor.
\(R=\cfrac { V }{ I } \)

Peltier discovered that, when an electric current is passed through a circuit of a thermocouple heat is evolved at one junction and absorbed at the other junction. This is known as Peltier effect.

The internal resistance of a cell is the resistance offered to the flow of current (by the electrolyte) inside the cell.

(i) Electrical power P = VI
(ii) Erectrical power \(P=V\left(\frac{V}{R}\right)=\frac{V^{2}}{R} \quad \therefore P=\frac{V^{2}}{R}\)
(iii) P = Iv = I(IR) = I²R
(iv) P = I²R

Electric power:
(i) The electric power P is the rate at which the electrical potential energy is delivered.
(ii) The electric power P is the rate at which the work is done.
\( P= \frac{d U}{d t} (or)= \frac{d W}{d t} (or)=VI(or)\frac{V^2}{R}\)
Unit: watt (W)
Electric energy:
(i) The electric energy is the product of power (P) and duration of the time (t) when electric energy is delivered.
(ii) E = Pt
Unit: watt-hour (Wh)

Microscopic form of ohm's law is 
\(\vec{J}=\sigma\vec {E}\) 
\(J=\frac{ne^2\tau}{m}\vec{E}\)
\(\frac{e \tau}{m} \rightarrow Drift \ velocity \ v_d\)
where \(\vec{J} \) - current density
\(\sigma \)  - conductivity
\(\vec{E} \) - Electric field