Why is current equal in 2 resistors of same area and length connected in series even if the drift speed in resistor of more resistivity is less and current is neAv and electric field inside both is same as assured by the battery?

To understand why the current is the same in two resistors connected in series, even when one has a higher resistivity and a lower drift speed, we need to delve into some fundamental concepts of electricity and how resistors behave in a circuit.

The Basics of Series Circuits

In a series circuit, components are connected end-to-end, forming a single path for current to flow. This means that the same amount of charge must pass through each component in a given time. Therefore, the current (I) remains constant throughout the circuit.

Current and Charge Flow

Current is defined as the rate of flow of charge. Mathematically, it is expressed as:

• I = Q/t

Where I is the current, Q is the charge, and t is time. Since the same charge flows through both resistors in series, the current must be the same, regardless of the resistivity of the materials.

Understanding Drift Velocity and Resistivity

Drift velocity (v_d) is the average velocity that a charge carrier, such as an electron, attains due to an electric field. It is influenced by the material's resistivity (ρ) and the electric field (E) applied across the resistor. The relationship can be expressed as:

• v_d = μE

Where μ is the mobility of the charge carriers. Higher resistivity means lower mobility, which can lead to a lower drift velocity for the same electric field. However, this does not affect the overall current in a series circuit.

Ohm's Law and Resistance

Ohm's Law states that:

• V = IR

Where V is the voltage across the resistor, I is the current, and R is the resistance. In a series circuit, the total voltage supplied by the battery is divided among the resistors based on their resistances. The current remains constant, but the voltage drop across each resistor varies according to its resistance.

Example for Clarity

Consider two resistors, R1 and R2, connected in series. Let’s say R1 has a resistivity of 1 ohm and R2 has a resistivity of 2 ohms. If a voltage of 12 volts is applied across the series combination, the total resistance (R_total) is:

• R_total = R1 + R2 = 1Ω + 2Ω = 3Ω

Using Ohm's Law, the total current (I) flowing through the circuit can be calculated as:

• I = V/R_total = 12V / 3Ω = 4A

This current of 4A flows through both resistors, even though R2 has a higher resistivity and a lower drift velocity. The electric field inside both resistors is the same because they are connected in series and the battery maintains a constant voltage across the entire circuit.

Final Thoughts

In summary, the current remains equal in resistors connected in series due to the nature of electric charge flow and the definition of current. While drift velocity and resistivity affect how quickly charge carriers move through a material, they do not change the fact that the same amount of charge must pass through each resistor in a series configuration. Thus, the current remains constant, ensuring that both resistors experience the same current despite their differing properties.