Electromagnetic Induction

A wire in the form of a circular loop of one turn carrying a current produces a magnetic field B at the centre. If the same wire is looped into a coil of two turns and carries the same current, the new value of magnetic induction at the centre is

The magnetic field at its center B = μ₀I/2πr

Thus the length of the wire L = 2πr

But the same wire is looped into a coil of radius r₂ of two turns

r₂ = r/2

The magnetic field at its center B = (μ₀I/2πr₂) x n = [μ₀I/2π(r/2)] x 2 = 4 x μ₀I/2πr =4B

B = (2) B = 4B

Magnetic field, also called a vector field, represents the magnetic influence on moving electric charges, magnetic materials, and electric currents. The magnetic fields force moving electrically charged particles in a circular or helical path and the charged particles experience a force perpendicular to their own velocity and to the magnetic field. Magnetic field can be expressed as the area around a magnet wherein the effect of magnetism is felt.

• The symbol of the Magnetic Field can be denoted by B or H. It is denoted mathematically by quantities known as vectors which have direction and magnitude both.
• Two different vectors help represent magnetic field: Magnetic flux density (or magnetic induction) and Magnetic field strength (or magnetic field intensity) each symbolized by B and H respectively.
• The unit of Magnetic Field is Telsa and its base unit is (Newton.Second)/Coulomb.
• Magnetic field lines are known to not cross one another. In fact, magnetic lines form closed loops, beginning from the north pole and ending at the south pole.
• The density of the field lines generally indicates the strength of the field.

Discover more from this chapter: Electromagnetic Induction

Let's analyze each field pattern against the properties of electric and magnetic fields.

(A) Field lines radiating outwards from a central point.

Electric Field: This is a valid representation of the electric field lines originating from a positive point charge.

Magnetic Field: This is NOT a typical valid representation of a fundamental magnetic field source. Magnetic fields do not originate from a point in the same way electric fields originate from point charges (magnetic monopoles are not observed).

(B) Field lines converging inwards towards a central point.

Electric Field: This is a valid representation of the electric field lines terminating at a negative point charge.

Magnetic Field: Similar to (A), this is NOT a typical valid representation of a fundamental magnetic field source. Magnetic fields do not terminate at a point in the same way electric fields terminate at point charges.

(D) Concentric circles.

Electric Field: Electric field lines cannot form closed loops in electrostatics. Electrostatic fields are conservative. Thus, this pattern is NOT valid for an electric field under static conditions.

Magnetic Field: This is a valid representation of the magnetic field lines around a long, straight current-carrying wire. According to Ampère's Law, the magnetic field lines form concentric circles around the wire.

(C) Field lines forming loops between two points (like a dipole).

Electric Field: This pattern can represent the electric field of an electric dipole (two opposite charges). Electric field lines originate from the positive charge and terminate at the negative charge, forming curved paths between them. While not strictly closed loops in the magnetic sense, the overall shape is dipole-like.

Magnetic Field: This pattern is a valid representation of the magnetic field of a magnetic dipole, such as a bar magnet or a current loop. Magnetic field lines emerge from the North pole, curve around, and enter the South pole, forming closed loops that continue inside the magnet.

Considering the question "Which of the field pattern given below is valid for electric field as well as for magnetic field?", we need to find a pattern that can represent both, even if not perfectly in every detail for both types of fields.

Option (C) represents a dipole field. Dipole fields are fundamental in both electrostatics (electric dipole) and magnetostatics (magnetic dipole). While electric field lines do not form closed loops in the same way as magnetic field lines of a magnet, the overall dipole pattern is a common and recognizable representation in both contexts.

Final Answer: The final answer is

Step 1: Consider the force on a stationary electron in a magnetic field.

The Lorentz force on a charge moving with velocity in a magnetic field is .

Step 2: Apply the condition of a stationary electron.

For a stationary electron, the velocity .

Therefore, the magnetic force .

Step 3: Determine the motion of the electron.

Since the net magnetic force on a stationary electron is zero (and assuming no other forces are considered in this simplified scenario), the electron will remain stationary.

The correct answer is (C) Remains stationary.

Step 1: Recall the relationship between induced electric field and rotation

When a conducting rod rotates about an axis with constant angular velocity, an electric field is induced in the rod due to the motion of charges in the conductor. This is a result of the Lorentz force acting on the free charges within the conductor. The key equation here is the relationship between the induced electric field and the velocity of the charges at a given point.

Step 2: Use the concept of induced electric field

The velocity of a point on the rod at a distance from the axis of rotation is given by:

where:
- is the angular velocity,
- is the distance from the axis of rotation.

The induced electric field in the rod at a distance from the axis of rotation is given by the formula for the electric field in a rotating conductor:

where:
- is the velocity of the point at distance , which is ,
- is the magnetic field generated by the rotating conductor,
- is the length of the rod.

Step 3: Apply the given values

We are given that the rod is rotating about end A with a constant angular velocity . The induced electric field will depend on the rotation of the rod, the magnetic field generated, and the distance from the axis.

Since the magnetic field generated by the rotating conductor will be proportional to the distance from the axis, we find the relation for the electric field. After substituting the necessary terms, we arrive at:

Final Answer:
The electric field at a distance from the axis of rotation is , which matches option (A).

Given: Mass per unit length of the metallic rod is

Mass per unit length .

Let be the current flowing through the rod. For equilibrium, the forces acting on the rod balance each other:

Solving for the current :

Substituting the known values:

Therefore, the current flowing through the metallic rod is .

A fully charged capacitor with an initial charge is connected to a coil of self-inductance at time . We want to find the time at which the energy stored in the electric field of the capacitor is equal to the energy stored in the magnetic field of the inductor.

In an LC circuit, the charge on the capacitor oscillates sinusoidally. The charge on the capacitor as a function of time is given by:

Where is the angular frequency of oscillation, and .

The energy stored in the capacitor at time is:

The current in the inductor is the derivative of the charge with respect to time:

The energy stored in the inductor at time is:

Since , then , and we can rewrite the inductor energy as:

We want to find the time when . Therefore:

This implies , which means , or .

The first positive solution for is . Therefore:

Therefore, the time at which the energy is stored equally between the electric field and the magnetic field is:

Answer: The correct statement is: (A) Fields are useful for understanding forces acting through a distance.

Explanation:

Fields are an essential concept in physics that help describe how forces can act through a distance without direct contact. There are two primary types of fields that describe forces acting at a distance:

• Electric fields: These are created by electric charges and describe the force that a charge exerts on other charges in its vicinity.
• Gravitational fields: These are created by masses and describe the force that a mass exerts on other masses in its vicinity.

Both fields act through space and influence other objects within their reach, which is why we can observe forces at a distance, such as gravitational attraction between planets or the electric force between charged particles.

Therefore, the correct answer is: (A) Fields are useful for understanding forces acting through a distance.

Electric Field:

Electric Potential:

Therefore, the correct option is: D.

Using the equation of motion, calculate the kinetic energy of a particle moving with constant acceleration, where and the energy involved is .

Step 1: Start with the equation of motion:

Rearrange the equation to solve for :

Step 2: Kinetic energy (K) of the particle:

The kinetic energy is given by:

Substitute the value of from earlier:

Step 3: Simplify the equation:

Therefore, the correct expression for the kinetic energy is: , which corresponds to option (C) .

Answer: The radiation that is deflected by an electric field is α-particles (option C).

Explanation:

• α-particles are positively charged particles consisting of two protons and two neutrons. This gives them a positive charge, and when they pass through an electric field, they experience a force due to their charge and can be deflected accordingly.
• γ-rays (option A) and X-rays (option B) are forms of electromagnetic radiation, which are not charged particles. Thus, they are not deflected by electric fields.
• Neutrons (option D) are neutral particles, meaning they have no net electric charge, so they are also not deflected by electric fields.

Therefore, the correct answer is: α-particles (option C).

Faraday's law of electromagnetic induction:

In this case, we are given:

• Flux density (B) = 1.0 Wb/m²
• Area of the coil (A) = 0.01 m²
• Number of turns (N) = 80
• Time taken (dt) = 0.2 seconds

The magnetic flux (Φ) passing through the coil is given by:

Now, we can calculate the rate of change of magnetic flux :

Since the coil is removed from the field, the final magnetic flux is zero ( ), and the initial magnetic flux ( ) is 0.01 Wb. Substituting these values:

Finally, substituting the values into the emf formula:

Therefore, the emf induced in the coil is 4 V. Option (B) is correct.

Given Information:
Initial current,
Final current,
Change in time,
Induced emf,

Step-by-Step Explanation:

Step 1: Using formula for induced emf due to self-induction:

Self-induced emf ( ) in a coil is given by the formula:

Where is the self-inductance, is the change in current, and is the time interval.

Step 2: Solve for self-inductance :

Calculate change in current, :

Substitute values:

Step 3: Convert inductance into millihenry (mH):

Final Conclusion:
The self-inductance of the coil is 100 mH.