Current Electricity Test - 48

The electrical resistance of an electrical conductor is the opposition to the passage of an electric current through that conductor.
The reasons for these changes in resistivity can be explained by considering the flow of current through the material. The flow of current is actually the movement of electrons from one atom to another under the influence of an electric field. Electrons are very small negatively charged particles and will be repelled by a negative electric charge and attracted by a positive electric charge. Therefore if an electric potential is applied across a conductor (positive at one end, negative at the other) electrons will "migrate" from atom to atom towards the positive terminal.
Only some electrons are free to migrate however. Others within each atom are held so tightly to their particular atom that even an electric field will not dislodge them. The current flowing in the material is therefore due to the movement of "free electrons" and the number of free electrons within any material compared with those tightly bound to their atoms is what governs whether a material is a good conductor (many free electrons) or a good insulator (hardly any free electrons).
The effect of heat on the atomic structure of a material is to make the atoms vibrate, and the higher the temperature the more violently the atoms vibrate.
In a conductor, which already has a large number of free electrons flowing through it, the vibration of the atoms causes many collisions between the free electrons and the captive electrons. Each collision uses up some energy from the free electron and is the basic cause of resistance. The more the atoms jostle around in the material the more collisions are caused and hence the greater the resistance to current flow.
In an insulator however, there is a slightly different situation. There are so few free electrons that hardly any current can flow. Almost all the electrons are tightly bound within their particular atom. Heating an insulating material vibrates the atoms, and if we heated sufficiently the atoms vibrate violently enough to actually shake some of their captive electrons free, creating free electrons to become carriers of current. 
Therefore at high temperatures the resistance of an insulator can fall, and in some insulating materials, quite dramatically.
In a material where the resistance increases with temperature, it is said that the material has a positive temperature coefficient.
When resistance falls with increase in temperature, the material is said to have a negative temperature coefficient.
In general, conductors have a positive temperature coefficient, while (at high temperatures) insulators have a negative temperature coefficient.

Components connected in series are connected along a single path, so the same current flows through all of the components. The current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component. In a series circuit, every device must function for the circuit to be complete. One bulb burning out in a series circuit breaks the circuit. A circuit composed solely of components connected in series is known as a series circuit.
The total resistance of resistors in series is equal to the sum of their individual resistances. That is, $${ R }_{ total }={ R }_{ 1 }+{ R }_{ 2 }$$. The current is given as $$I={ I }_{ 1 }={ I }_{ 2 }$$.
Below is the diagrammatic representation of 2 resistors $${ R }_{ 1 }\quad and\quad { R }_{ 2 }$$ connected in series.

Equivalent resistance is more than either of the two resistances when they are connected in series.Components connected in series are connected along a single path, so the same current flows through all of the components.  The current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component. In a series circuit, every device must function for the circuit to be complete. One bulb burning out in a series circuit breaks the circuit. A circuit composed solely of components connected in series is known as a series circuit.The total resistance of resistors in series is equal to the sum of their individual resistances. 

In the first case, there are 2 resistors of 2 ohms each connected in parallel. Hence, the emf of the cell is given as 1.8 volts.

Slope of the graph will give us reciprocal of resistance. Here resistance at temperature $$T_1$$ is greater than that of $$R_2$$. Since resistance of metallic wire is more at higher temperature than at lower temperature, hence $$T_1 >  T_2$$

Here $$R$$ and $$r$$ are in series so total resistance of closed loop circuit is $$R_t=R+r$$

We know that resistance of material depends upon temperature by this condition :  $$\displaystyle R_t = R_0 (1 + \alpha \Delta T)$$
for conductors $$\alpha$$ is positive in nature. Hence, resistance of a conductor increases with increase in temperature.

For balanced Wheatstone bridge which is shown in figure a, $$\dfrac{P}{Q}=\dfrac{S}{R}$$
If we interchange the cell and galvanometer then circuit becomes as shown in figure b.
and balanced condition, $$\dfrac{P}{S}=\dfrac{Q}{R} \Rightarrow \dfrac{P}{Q}=\dfrac{S}{R}$$
Thus, balanced point remains unchanged.

When the circuit is closed, the resulting current not only flows through the external circuit, but through the source (cell) itself. The cell have an internal resistance, which causes an internal voltage drop, slightly reducing the voltage across the terminals. The larger the current, the larger the internal voltage drop, and the lower the terminal voltage.
The current in the circuit is calculated from the formula $$I=\dfrac { \varepsilon  }{ R+r } $$ where,
$$\varepsilon $$- emf of the cell
R- external resistance
r- internal resistance of the cell.
When the cell is connected to the resistance of 1.9 ohms the current relation is given as 
That is, $$1.0=\dfrac { e }{ 1.9+r } .\quad That\quad is,\quad after\quad simplification,\quad e-r=1.9\quad -\quad eqn\quad 1$$.
When the cell is connected to the resistance of 3.9 ohms the current relation is given as 
That is, $$0.5=\dfrac { e }{ 3.9+r } .\quad That\quad is,\quad after\quad simplification,\quad e-0.5r=1.95\quad -\quad eqn\quad 2$$.
Solving the linear eqn 1 and eqn 2 for the variables e and r, we get e=2.0 V.
Substituting the value of e in eqn 1, the variable r is determined to be 0.1 ohms.
Hence, the emf and the internal resistance of the cell are 2.0 V and 0.1 ohms respectively.