**Formulas of Chapter Electricity and Their Derivations**

Electricity is a crucial topic in science that deals with the flow of electric charge. It is important to understand the formulas and their derivations to solve various numerical problems related to electricity. Here, we will discuss the key formulas and their derivations in detail.

**1. Ohm's Law:**

Ohm's Law states that the current flowing through a conductor is directly proportional to the potential difference across it, provided the temperature and other physical conditions remain constant. Mathematically, it can be expressed as:

I = V/R

Where:
- I is the current flowing through the conductor (in amperes, A)
- V is the potential difference across the conductor (in volts, V)
- R is the resistance of the conductor (in ohms, Ω)

**Derivation of Ohm's Law:**

Ohm's Law can be derived using the concept of drift velocity and the relationship between current and charge. When a potential difference is applied across a conductor, it creates an electric field that accelerates the free electrons. These accelerated electrons collide with the atoms in the conductor, resulting in a drift velocity.

The current flowing through the conductor is the rate of flow of charge, given by:

I = q/t

Where:
- q is the charge flowing through the conductor (in coulombs, C)
- t is the time taken (in seconds, s)

Using the relationship between charge and potential difference (q = V × C), and substituting the expression for charge in the above equation, we get:

I = V × C/t

By rearranging the equation, we obtain:

I = V × (1/t)

Since drift velocity (v_d) = l/t (where l is the length of the conductor), we can substitute (1/t) with (v_d/l) to get:

I = V × (v_d/l)

The drift velocity (v_d) is directly proportional to the electric field (E) applied across the conductor, and the electric field is given by E = V/l. Substituting these values, we get:

I = (V/l) × (v_d)

Let (V/l) be represented by constant R (resistance), and (v_d) by constant I (current), the equation becomes:

I = V/R

Hence, Ohm's Law is derived.

**2. Electric Power:**

The electric power consumed by an electrical device can be calculated using the formula:

P = VI

Where:
- P is the electric power consumed (in watts, W)
- V is the potential difference across the device (in volts, V)
- I is the current flowing through the device (in amperes, A)

**Derivation of Electric Power:**

The electric power can be derived by substituting Ohm's Law (I = V/R) into the formula for power:

P = VI = (V/R) × V = V^2/R

Hence, the derived formula for electric power is P = V^2/R.

These are the two key formulas in the chapter on electricity, along with their derivations. Understanding these formulas and their derivations will help in solving numerical problems related to electricity and analyzing various electrical circuits.

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