Question

The forces existing between two parallel current carrying conductors is \(\mathrm{F}\). If the current in each conductor is doubled, then the value of force will be (a) \(2 \mathrm{~F}\) (b) \(4 \mathrm{~F}\) (c) \(5 \mathrm{~F}\) (d) \((\mathrm{F} / 2)\)

Short answer

The short answer to the question is: If the current in each conductor is doubled, the new value of the force between the two parallel current carrying conductors will be: \( 4 \mathrm{~F} \)

Step-by-step solution

Write the formula for the force between parallel conductors

The formula for the force per unit length (f) between two long parallel conductors with currents I₁ and I₂ separated by a distance d is given by: \( f = \dfrac{\mu_0 I_1 I_2}{2 \pi d} \) Where: \( \mu_0 \) is the permeability of free space (approximately \( 4 \pi × 10^{−7} \) T⋅m/A), I₁ and I₂ are the currents in the conductors, and d is the distance between the conductors.

State the given values and the values of the doubled current

The given force between the conductors is F. Let's denote the original current in each of the conductors as \( I \). When the currents are doubled, the new current values will be \( 2I \) in each conductor.

Write the formula for the force after doubling the current

Now, to find the force between the conductors after the currents are doubled, we can use the same formula, substituting the doubled current values \( 2I \) for I₁ and I₂: \( f' = \dfrac{\mu_0 (2I) (2I)}{2 \pi d} \)

Relate the original force with the new force to find the change in force.

To find the change in force, we need to relate the original force (F) to the new force (f'). We can do this by dividing the second equation by the first equation: \( \frac{f'}{f} = \frac{\dfrac{\mu_0 (2I) (2I)}{2 \pi d}}{\dfrac{\mu_0 I I}{2 \pi d}} \) Now, let's simplify the equation to find the change in force: \( \Rightarrow \frac{f'}{f} = \frac{(2I)(2I)}{I I} \) \( \Rightarrow \frac{f'}{f} = 4 \) This tells us that the new force \( f' \) is 4 times the original force (F).

Select the correct answer from the options given.

The correct answer for the value of the force when the currents in both conductors are doubled is: (b) \( 4 F \)

Key concepts

Magnetic fields and forces

When currents pass through wires, they create magnetic fields around them. This can be imagined like invisible rings, circling around each wire. These fields are very important because they allow currents in different wires to interact with each other.

Now, when two wires carrying current are placed parallel to each other, the magnetic fields they produce overlap and interact. This interaction results in a force between the wires. If the currents flow in the same direction, the wires attract each other; if the currents flow in opposite directions, they repel each other. It's like magnets where like poles repel and unlike poles attract.

The strength of the force depends on several things:

• The amount of current flowing in the wires.
• The distance between the wires.
• The medium around the wires, though usually the wires are in air or vacuum.

Ampère's force law

Ampère's force law describes how two parallel currents affect each other. It's a fundamental principle in physics that predicts the force based on the values of currents and their proximity.

The formula derived from Ampère's force law is:\[ f = \dfrac{\mu_0 I_1 I_2}{2 \pi d} \]Where:

• \( \mu_0 \) stands for the permeability of free space, a constant that measures how much resistance is there to the magnetic field in a vacuum.
• \( I_1 \) and \( I_2 \) are the currents flowing through the wires.
• \( d \) is the distance between the two wires.

This law shows that the force is directly proportional to the product of the currents and inversely proportional to the distance. It's crucial because it helps us predict and calculate the force between two current-carrying conductors.

Electromagnetism

Electromagnetism is the branch of physics that studies the interaction between electric currents and magnetic fields. It combines electricity and magnetism, showing us how they are part of the same force.

A primary example of electromagnetism is how passing an electric current through a wire creates a magnetic field. This is the foundation of how electromagnets work. Unlike permanent magnets, whose magnetic field is always present, electromagnets can be turned on and off with the current.

Electromagnetism is a crucial part of many technological advancements. Devices like electric motors, generators, and transformers rely on this principle. Understanding electromagnetism allows scientists and engineers to design systems that convert electricity into motion or vice versa.

Current electricity

Current electricity refers to the flow of electric charges through a conductor. It's what powers our electronic devices, lights up our homes, and allows us to use all sorts of appliances.

The flow of electricity is driven by voltage, much like how water flows through a pipe because of pressure. In wires, this electric flow is made up of moving electrons, and it happens quite rapidly, allowing devices to work almost instantly when turned on.

Understanding current electricity is fundamental to grasping how electrical circuits work. Key factors include:

• Voltage (V): The driving force pushing the current through a circuit.
• Current (I): The rate at which charge flows through a point in the circuit.
• Resistance (R): A measure of how much a material resists the flow of current.

Electricity can be direct current (DC) where electrons flow in one direction, or alternating current (AC) where they reverse direction at regular intervals, common in power systems.