Classify the following elements into metals, non–metals and metalloid: C, Mg, Si, S, Hg, As.

Metals	Non – Metals	Metalloids
Mg	C	Si
Hg	S	As

Resistance connected in parallel

Resistors connected in parallel: If the numbers of resistance are connected between two common points, such that the potential difference across each resistance is the same, then the arrangement is called resistance in parallel.

Three resistances R₁, R₂ and R₃ are connected in parallel between the points A and B. Let R_p be the effective resistance in the circuit.

A Cell E, Key K and the ammeters A are also connected with resistances.

Let the current passing through R₁ be I₁, R₂ be I₂, and R₃ be I₃ and that of R be I.

Derivation of Equivalent Resistance in a Parallel Circuit

The image provided outlines the mathematical derivation for calculating the total equivalent resistance ($R_p$) when three resistors ($R_1$, $R_2$, and $R_3$) are connected in parallel. Below is a detailed, step-by-step breakdown of the physics principles shown.

Step 1: The Current Rule in Parallel Circuits

In a parallel circuit, the total current flowing from the source divides into the different parallel branches. Therefore, the total current ($I$) is the sum of the currents in each individual branch.

$$I = I_1 + I_2 + I_3 \quad \text{--- eq no (1)}$$

Step 2: Applying Ohm's Law

According to Ohm's law, Current ($I$) is equal to Voltage ($V$) divided by Resistance ($R$). A fundamental characteristic of parallel circuits is that the voltage ($V$) remains constant across all branches.

Applying this to each resistor gives us their individual currents:

$$I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3}$$

For the whole circuit, utilizing the total equivalent resistance ($R_p$), the total current is:

$$I = \frac{V}{R_p} \quad \text{--- eq no (2)}$$

Step 3: Substitution

By substituting the Ohm's law relationships from equation (2) into the total current equation (1), we replace the $I$ variables with their $\frac{V}{R}$ equivalents.

$$\frac{V}{R_p} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$$

Step 4: Factoring Out Common Variables

Because the voltage ($V$) is identical in every term on the right side of the equation, it can be factored out as a common multiplier.

$$\frac{V}{R_p} = V \left[ \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right]$$

Step 5: Final Formula for Parallel Resistance

Finally, dividing both sides by the common voltage $V$ cancels it out completely. This results in the standard formula used to calculate the equivalent resistance of resistors arranged in parallel.

$$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$

Conclusion: If the resistors are connected in parallel then:

1. The sum of reciprocals of the individual resistance is equal to the reciprocal of equivalent resistance.
2. The current in various resistors are inversely proportional to the resistances (higher is the resistance lower is the current through it). However the total current is the sum of the currents flowing in the different branches.
3. The voltage (potential difference) across each resistor is same.
4. The effective resistance of the parallel combination is less than the individual resistance in the combination.
5. This combination is used to decrease resistance in the circuit.

Find the expression for the resistors connected in series

Resistors connected in series: - If the number of resistances are connected one after another in such a way that the same current flows through each resistance, then the arrangement is called resistance in series.

Let R₁, R₂ and R₃ be three resistances connected in a series combination and let R_S be their effective resistance in the circuit.

Let V₁, V₂ and V₃ be the P.D. across resistances R₁, R₂ and R₃ respectively. Let ‘V’ be the P.D. of the cell. Let ‘I’ be the current flow through each resistance.

According to Ohm’s Law,

Conclusion: If the resistors are connected in series then:

1. In a circuit the current is the same in every part of the circuit.
2. The resistance of the combination of resistors is equal to the sum of the individual resistors.
3. The total voltage across the combination is equal to the sum of the voltage drop across the separate resistors.
4. The effective resistance in a series combination is greater than the individual resistances.
5. This combination is used to increase resistance in a circuit.

Joule’s law

- Joule’s law can be stated as the quantity of heat generated (H) in a conductor of Resistance (R), when a current (I) flows through it for a time (t) is directly proportional to:

i.                     The square of the current.

ii.                    The resistance of the conductor, and

iii.                  The time for which the current flows.

Using Ohm’s Law we can write

H	=	(I2 Rt) cal	=	(V2 t) cal	=	VIt cal
		4.18		4.18R		4.18

Ohm’s law

 Ohm’s law states that the electric current flowing in a metallic conductor is directly proportional to the potential difference across its terminals, provided physical conditions of the conductor such as length, area of cross section, temperature and material remain constant.

If I is the current and V is the potenial difference across the ends of a conductor then, V ∝ I

∴ V / I = R  (Where ‘R’ is constant)

∴ V = IR

R is the resistance which is constant for given conductor. The SI unit of resistance is ohm ( Ω )

High resistance and low resistance.

High resistance	Low resistance
A high resistance indicates a material that hardly allows the movement of electrons. It is due to the less number of free flowing electrons in the outer most orbit of an element. Substances with infinitely high electrical resistance are insulators. High resistance provides low conductivity.	A low resistance indicates a material that readily allows the movement of electrons. It is due to large number of electrons in the outer most orbit of an element. Substances with low electrical resistances are good conductors. Low resistance provides high conductivity.

Resistance and resistivity.

Resistance	Resistivity
The property of the conductor due to which it opposes a flow of current through it is called resistance. The SI unit of resistance is Ohm ( Ω ) The resistance of a conductor depends on its length and area of cross section.	The resistivity of a conductor is the resistance of a conductor of unit length and unit area of cross section. The SI unit of resistivity is Ohm-metre ( Ωm)  The resistivity of a conductor does not depend on its length and area of cross section.

Conductors and insulators.

Conductors	Insulators
·         Those substances through which electricity can flow are called conductors. ·         Electrical resistances of conductors are very low. ·         They contain large number of free electrons. ·         Generally metals are conductors. E.g. silver, copper, aluminium	·         Those substances through which electricity cannot flow are called insulators. ·         Electrical resistances of insulators are infinitely very high. ·         They do not contain free electrons. ·         Generally non – metals are insulators. E.g. wood, rubber, plastic

Resistances in series and parallel.

Resistance in series

Resistance in parallel

1.       If a number of resistances are connected in such a way that the same current flows through each resistance, then the arrangement is called resistances in series. 2.       The effective resistance is a series combination is greater than the individual resistances. 3.       This combination is used to increase resistance in a circuit. 4.       This combination decreases the current in the circuit.

1.       If a number of resistances are connected between two common points such that the potential difference across each is the same then that arrangement is called resistances in parallel. 2.       The effective resistance of the combination is less than the individual resistances. 3.       This combination is used to decrease resistance in the circuit. 4.       This combination increases the current in the circuit.