Current Electricity

(a) Current Density

Current density is the current per unit area and is a vector. It is given by:

Derivation

The drift velocity is related to the electric field as:

Where is the charge of an electron, is the relaxation time, and is the mass of the electron.

The current density is given by the product of the number of charge carriers , the charge of the carriers , and their drift velocity :

Simplifying this, we get:

Thus, the conductivity is:

(b) Wheatstone Bridge

The Wheatstone bridge is used to measure an unknown resistance. It is balanced when the following condition is met:

In this case, there is no current flowing through the galvanometer, and the bridge is considered balanced.

1. Resistance of a Conductor:

The resistance of a conductor is given by the formula:

Where:

• is the resistivity of the material.
• is the length of the conductor.
• is the cross-sectional area of the conductor.

When the conductor is stretched, its length increases, and the cross-sectional area decreases (since the volume of the conductor remains constant during stretching). If the length of the conductor is doubled to , the cross-sectional area becomes half of its original value.

2. Relation Between Initial and Final Resistance ( and ):

Let the initial length be and the initial cross-sectional area be , and let the final length be and the final cross-sectional area be . The resistance after stretching the conductor is given by:

Thus, the ratio of the final resistance to the initial resistance is:

Therefore, the relation between the final and initial resistance is:

3. Drift Velocity:

The drift velocity of the electrons is given by the equation:

Where:

• is the current flowing through the conductor.
• is the number of free electrons per unit volume of the material.
• is the charge of an electron.
• is the cross-sectional area of the conductor.

Since the current is constant (because the emf and resistance are constant), and and are constant, we observe that the drift velocity is inversely proportional to the cross-sectional area of the conductor:

When the length of the conductor is doubled, the cross-sectional area is halved. Therefore, the drift velocity after stretching becomes:

4. Conclusion:

• Relation between final and initial resistance: .
• Relation between final and initial drift velocity: .

To find the resistance of wire Q, we begin by using the formula for resistance: where is the resistivity, is the length, and is the cross-sectional area.

For wire P, the resistance is given as .

Since wire Q has twice the diameter of wire P, its cross-sectional area is four times that of P (because area ). Thus, .

Wire Q also has half the length of wire P, so .

The resistance of wire Q, , is: .

Therefore, the resistance of wire Q is . The correct answer is .

The drift velocity is inversely proportional to the area of cross-section of the conductor, and directly proportional to the current and length.

Since the conductors are in parallel, the voltage is the same across both. Hence, the ratio of drift velocities is: Thus, the correct ratio is , corresponding to option (3).

Step 1 — Write branch currents (node form)

Take the potential of Q as reference (0). Let Vp = V_P − V_Q be the unknown potential difference between P and Q that we must find. Assume each battery has its positive terminal connected to the left node P (as drawn). For a branch with emf E and series resistance r (battery on left, resistor on right), the steady current from P → Q is (Derivation: the node between battery and resistor has potential V_p − E, and current through resistor = (V_p − E − V_Q)/r.)

Step 2 — Apply KCL at node P

Sum of currents (from P to Q) in all branches must be zero (no external injection at P): In steady state I_mid = 0 (capacitor open), so

Step 3 — Solve algebraically for V_p

Multiply by 2R to clear denominators: Combine terms: Therefore

Step 4 — Voltage across the capacitor

Middle branch (left-to-right): P → battery (emf V) → capacitor → Q. Let the node between the battery and capacitor be A. Then The capacitor is between A and Q, so the voltage across capacitor (left plate minus right plate) is Substitute V_p = 4V/3: Hence

Step 5 — Quick check via branch currents

Compute top-branch current: Bottom-branch current: So the two resistive branch currents are equal in magnitude and opposite in sign (their sum = 0), confirming the KCL balance. The middle branch carries no steady current.

Final answers (concise)

(i) Potential difference between P and Q: V_PQ = 4V/3.
(ii) Potential difference (magnitude) across capacitor C: V_C = V/3.

Given:

• Same material and radius resistivity and cross-sectional area are same
• Length ratio: .
• Total current from battery: .
• Connection: Parallel.

Step 1: Relation between resistance and length

For a wire: Since and are the same, the resistances are in the same ratio as lengths: Let and .

Step 2: Current division in parallel

In parallel: Substituting : Simplify: Similarly:

Final Answer:
through the shorter wire ( ),
through the longer wire ( ).

Note: In parallel, the wire with smaller resistance (shorter length) carries more current.

(i) Relation between R′ and R:

The resistance R of a conductor is given by:

When the length is increased to 2l, the cross-sectional area A decreases to A/2 (assuming the volume remains constant). Thus:

So,

(ii) Relation between v′_d and v_d:

Drift velocity v_d is given by:

When the length is doubled, the current I remains the same (since the cell is ideal), but the cross-sectional area A is halved. Thus:

So, .

Step 1: Write the equation for the first scenario (current = 2A)

The potential difference across the rheostat is given as:

V₁ = E - (2 × r) = 5V

Therefore, we have the equation:

E - 2r = 5

Step 2: Write the equation for the second scenario (current = 4A)

The potential difference across the rheostat becomes:

V₂ = E - (4 × r) = 4V

Thus, we have the equation:

E - 4r = 4

Step 3: Solve the system of equations

We now have the following system of equations:

 E - 2r = 5 E - 4r = 4 

By subtracting the second equation from the first, we get:

2r = 1

So, r = 1/2 Ω

Step 4: Substitute the value of r into one of the equations

Substitute r = 1/2 Ω into the first equation:

E - 1 = 5

Therefore, E = 6V.

Final Answer:

EMF of the battery: E = 6V

Internal resistance of the battery: r = 1/2 Ω

When identical cells are connected in series, the total effective e.m.f. is: If two cells are connected in reverse polarity, their individual e.m.f. will subtract from the total e.m.f.: However, the potential difference across a single reversed cell ‘X’ is equal to the sum of the e.m.f. of the reversed cells, which is: Thus, the potential difference across cell ‘X’ is .

Let the equivalent EMF of the two cells be , and the equivalent internal resistance be . Since the cells are connected in parallel, the terminal voltages across both cells are equal. We can write:

Where is the terminal voltage across both cells, and are the currents through the two cells, and and are their respective internal resistances.

The total current supplied by the equivalent cell is the sum of the currents from the two cells:

Now, expressing the currents and in terms of the terminal voltage , we get:

Thus, the total current is:

Now, rearrange the equation to get the total current in terms of the terminal voltage :

Now, using Ohm’s law for the equivalent cell, we know that:

Substituting the expression for into this equation:

Now solve for and , we get:

1. Equivalent EMF:

The equivalent EMF is given by:

2. Equivalent Internal Resistance:

The equivalent internal resistance is given by:

Thus, the equivalent internal resistance of the two cells connected in parallel is .

Summary of the results:

(a) EMF and Terminal Voltage

EMF (Electromotive Force):
EMF is the maximum potential difference between the terminals of a cell when no current is being drawn from it. It represents the total energy supplied per unit charge by the cell.

Terminal Voltage:
Terminal voltage is the potential difference between the terminals of the cell when it is supplying current. Due to internal resistance , some voltage is lost inside the cell. Hence:

Difference:

(b) Derivation for Two Cells in Parallel

Given: Two cells of EMFs and , and internal resistances and , connected in parallel.

Objective: Derive the expression for equivalent EMF and equivalent internal resistance .

Solution:
Since the cells are connected in parallel, their terminal voltages must be equal. Let:

Let the total current be , and for the equivalent cell:

From the current expressions:

Total current becomes:

Substitute into :

Solve for and , and we get:

Equivalent EMF:

Equivalent Internal Resistance:

Calculation of Internal Resistance

Given:

• Current
• Current

Step 1: Cross Multiply the Equations

From the given current equations, we can cross-multiply:

Step 2: Equating the Two Expressions

Equating the two equations for , we get:

Step 3: Expanding and Simplifying

Expanding both sides:

Rearranging the terms:

Factor out :

Step 4: Solving for

Solving for , we get:

Final Answer:

The internal resistance is given by:

Expression for Current, Terminal Voltage, and Internal Resistance

(I) Expression for Current and Maximum Current

By Ohm's Law, the total resistance in the circuit is , where is the external resistance and is the internal resistance.

The current in the circuit is given by:

where is the electromotive force (EMF) of the battery.

The maximum current occurs when the external resistance , and is given by:

(II) Terminal Voltage and its Maximum Value

The terminal voltage is the potential difference across the external resistance, and is given by:

The maximum terminal voltage occurs when the current is minimum (i.e., when , meaning no current flows). In this case, the terminal voltage becomes equal to the EMF of the battery:

(III) Expression for Internal Resistance in Terms of

Given the following expressions for currents and with resistances and , respectively:

Cross-multiply the above equations: Equating the two expressions: Expanding and simplifying: Factor out : Solving for :

Final Answer:

• The expression for the internal resistance is:

Charge Calculation from Current-Time Graph

Given:

• Charge is the area under the current-time ( - ) graph.
• Formula for charge:

Interval for : From to , current is constant at 2 A

For this interval, the current is constant, so the area under the curve is simply a rectangle:

Interval for : From to , current increases linearly from 0 to 2 A

For this interval, the current-time graph forms a triangle. The area of a triangle is given by:

Substituting the values (base = 2 s, height = 2 A):

Comparison:

From the calculations above:

• Charge :
• Charge :

Thus, we have:

Derivation of Resistivity from Drift Velocity

Given:

• n = number density of free electrons (number of free electrons per unit volume)
• e = electronic charge
• m = mass of an electron
• = relaxation time (time between collisions of electrons)

Step 1: Drift Velocity of Electrons

When an electric field is applied to the conductor, the free electrons experience a force and accelerate, attaining an average drift velocity . The equation of motion for the drift velocity is given by:

Step 2: Current Density

The current density is given by the product of the number density of free electrons, the electron charge, and the drift velocity:

Substituting the expression for :

Step 3: Relation Between Current Density and Electric Field

The relationship between current density and the applied electric field is given by Ohm's Law:

where is the conductivity of the material. Comparing the two expressions for , we can equate the coefficients of to find the conductivity :

Step 4: Resistivity

Resistivity is the reciprocal of conductivity. Thus:

Final Answer:

The resistivity is given by:

To address the question, we need to understand the behavior of free electrons in a conductor when subjected to an electric field.

1. Understanding the Concepts:

- Free Electrons: In a conductor, free electrons are those that are not bound to a particular atom and can move freely through the material. These electrons are responsible for the flow of current when an electric field is applied.

- Drift Velocity: When an electric field is applied across a conductor, the free electrons experience a force that causes them to drift towards the higher potential. This movement of electrons is called the "drift velocity" and is generally very slow compared to the speed of the individual random motion of the electrons.

- Electric Potential: The electric potential is higher at the end of the conductor where the electric field is applied. Electrons, being negatively charged, move from regions of lower potential to higher potential under the influence of the electric field.

2. Do All Free Electrons Move in the Same Direction?

- While electrons drift from lower to higher potential, not all free electrons in the conductor are moving in the same direction at any given instant. The reason for this is the nature of the random thermal motion of the electrons. Each electron has its own random motion due to its thermal energy, which is much faster than the drift motion caused by the electric field.

- The overall drift velocity, which is the net movement of electrons due to the applied electric field, is very small and is in the direction from lower to higher potential (opposite to the electric field). However, the individual electrons still undergo random motion in all directions, which results in a random distribution of velocities.

3. Conclusion:

No, not all the free electrons move in the same direction. The electrons experience random motion due to thermal energy, but they drift overall in the direction opposite to the electric field due to the applied potential difference.

Final Answer:

While electrons drift in the direction from lower to higher potential, their random thermal motion means that not all of the free electrons are moving in the same direction at any given moment.

High Voltage Battery and Its Internal Resistance

A high voltage battery typically has a large internal resistance. This design choice serves several important purposes:

1. Limiting Current:

The internal resistance of the battery limits the amount of current that can be drawn from it. The current supplied by a battery is determined by the battery’s voltage and the total resistance in the circuit: where includes both the resistance of the circuit and the battery's internal resistance.

By having a large internal resistance, the current drawn from the battery is kept at a safe level. This is particularly important for high voltage batteries, as excessive current could lead to overheating, damage to the battery, or even dangerous conditions.

2. Preventing Overloading:

High voltage batteries are typically used to power devices that require a large voltage to operate. However, the power required by these devices is also dependent on the current drawn. If the battery had very low internal resistance, the current could easily exceed safe limits, especially if the external circuit’s resistance is low.

The large internal resistance of the battery ensures that the current remains within a safe range, preventing overloading of both the battery and any connected devices. This regulation helps prevent excessive heat generation and reduces the risk of battery failure.

3. Heat Generation and Damage Prevention:

When excessive current flows through a battery or a circuit, it leads to power dissipation in the form of heat. This is governed by the formula: where is the power dissipated as heat. High current can lead to significant heat generation, which can damage the battery and other connected components.

By using a high internal resistance, the battery limits the current, thereby preventing the buildup of excessive heat and avoiding potential damage to the battery or the devices it powers.

Conclusion:

In summary, high voltage batteries are designed with large internal resistance to limit the current drawn, preventing overloading, excessive heat generation, and potential damage. This ensures the battery operates efficiently and safely, protecting both the battery and the connected devices.

The statement "A low voltage battery from which high current is required must have low internal resistance" is justified by considering the following factors:

1. Ohm’s Law and Internal Resistance:

According to Ohm's law, the total current ( ) drawn from a battery is determined by the applied voltage ( ) and the total resistance in the circuit, which includes the internal resistance of the battery ( ) and the external resistance ( ):

If a high current is required, the current will be maximized when the total resistance in the circuit is minimized. This means that for a low voltage battery, if we need a high current, we must minimize the internal resistance of the battery.

2. Impact of High Internal Resistance:

When a battery has a high internal resistance, a significant portion of the battery's voltage is "lost" within the battery itself. This is because the internal resistance causes a voltage drop ( ) inside the battery, which reduces the voltage available for the external circuit.

• If the internal resistance is too high, it limits the amount of current that can flow out of the battery, as much of the voltage is dropped across the internal resistance.
• This results in inefficient performance, especially when high current is required by the external load.

3. Low Internal Resistance for High Current:

To ensure that a low voltage battery can supply a high current, the internal resistance must be kept low so that a minimal voltage is lost inside the battery. A low internal resistance allows most of the battery's voltage to be applied across the external circuit, allowing for higher current flow as required by the load.

4. Practical Example:

Consider a low voltage battery, such as a 1.5 V AA battery, that needs to supply current to a motor. If the battery has a high internal resistance, much of the 1.5 V would be dropped inside the battery, leaving little voltage for the motor, which would result in low current and poor motor performance. However, if the battery has low internal resistance, it can supply the full 1.5 V (minus a small drop) to the motor, resulting in a higher current and better performance.

5. Conclusion:

For a low voltage battery that needs to supply high current, it is essential to have low internal resistance. This ensures that most of the voltage from the battery is applied to the external circuit, allowing for higher current flow and efficient operation of the load.

Definition of Resistivity: Resistivity of a material is a property that quantifies how strongly the material resists the flow of electric current. It is defined as: Where:
is the resistance of the conductor,
is the cross-sectional area,
is the length of the conductor.

Dependence of Resistivity on Temperature: The resistivity of most conductors increases with an increase in temperature. This is because the atoms in the conductor vibrate more at higher temperatures, impeding the flow of electrons. The temperature dependence of resistivity is given by: Where: - is the resistivity at temperature , - is the resistivity at a reference temperature , - is the temperature coefficient of resistivity.

Plot of Resistivity of Copper: The plot of resistivity of copper with respect to temperature shows a linear increase with temperature in the range commonly encountered.

(i) When a conductor is stretched, its resistance changes because resistance is proportional to the length of the conductor. The resistance is given by: where is the length and is the cross-sectional area. When the length is doubled, the resistance also doubles: (ii) The drift velocity is inversely proportional to the length of the conductor (as the potential difference and electric field remain the same): Thus, the relations are: Thus, the final resistance is twice the initial resistance, and the final drift velocity is half the initial drift velocity.

(i) The equivalent emf of two batteries connected in parallel is given by: where , , , and . Substituting the values: (ii) The equivalent internal resistance of two batteries in parallel is given by: Substituting the values: (iii) The total resistance in the circuit is . The current drawn from the combination is given by Ohm's law: Thus, the current drawn from the combination is approximately .

Electron Motion in a Conductor

In a conductor, free electrons move randomly due to thermal motion. However, when an electric field is applied, the electrons experience a net drift in the direction opposite to the applied field (because electrons are negatively charged). Let’s break down the concept:

Random Motion and Net Drift

Free electrons in a conductor move in random directions due to their thermal motion. This random motion occurs at all times and is not influenced by an external electric field.

When an electric field is applied, the electrons experience a force and begin to drift in the direction opposite to the field. However, this drift is superimposed on their random thermal motion.

Thus, while there is a net drift of electrons in one direction (opposite to the electric field), the electrons still move randomly in all directions at any given moment.

Conclusion

In summary, although free electrons in a conductor have a net drift in the direction opposite to the applied electric field, they are still moving randomly in all directions due to thermal motion. Therefore, not all free electrons move in the same direction at any given moment.

Derivation of Ohm's Law and Condition for Its Breakdown

1. Derivation of Ohm's Law from

The equation is known as the electrical conductivity equation. Here:

• is the electric field in the material,
• is the current density (current per unit area),
• is the resistivity of the material.

From this equation, we can express the electric field in terms of current density:

Now, consider **Ohm's law**, which states that the current density is proportional to the electric field and the material's conductivity (the inverse of resistivity). So, we can write:

Since , we can substitute this into the above equation:

Rearranging the equation, we get:

This is exactly the form of the equation we started with, so we have derived Ohm's law from the equation .

2. Condition Under Which Ohm's Law Is Not Valid

Ohm's law assumes that the material has a constant resistivity and that the current is proportional to the applied voltage (i.e., linear response). However, there are conditions under which Ohm's law does not hold:

• Non-Linear Materials: If the resistivity changes with the applied electric field, temperature, or current density, the relationship between and becomes non-linear, and Ohm's law does not apply.
• High Electric Fields: In materials with very high electric fields (such as in semiconductors at high bias voltages), the current may no longer be proportional to the electric field, leading to a breakdown of Ohm's law.
• Temperature Dependence: In certain materials, particularly at very high temperatures or low temperatures, resistivity can vary significantly, leading to deviations from Ohm’s law.
• Superconductivity: In superconductors, the resistivity drops to zero below a certain temperature, and no electrical resistance is present, hence Ohm’s law does not hold.

Thus, Ohm’s law is not valid in situations where the material’s resistivity is not constant or when extreme conditions like high electric fields or temperatures cause a non-linear relationship between voltage and current.

The resistance of a wire is given by the formula:

where is the resistivity of the material, is the length of the wire, and is the cross-sectional area.

The cross-sectional area of a wire with radius is:

So, the resistance can be expressed as:

To achieve , the new resistance must be half of the original; hence:

Let us analyze the options:

• Using a wire of same radius and twice the length:
• Using a wire of same radius and half length:
• Using a wire of same length and twice the radius:
• Using a wire of same length and half the radius:

Therefore, the correct choice is: Using a wire of same radius and half length.