Magnetism And Matter

The correct answer is (D) :
Inside a hollow cylindrical conductor with uniform current distribution net magnetic field is zero in hollow space.
But outside the cylindrical conductor

The correct answer is (B) : 8 : 3

The correct answer is (A) :

Based on fundamental electrical engineering concepts:
- An AC generator operates on Electromagnetic Induction (II).
- A Transformer operates based on Mutual Inductance (IV).
- Resonance phenomenon occurs due to Presence of both L and C (I).
- Sharpness of Resonance is linked to Quality Factor(III).

Soft iron is used in electromagnets due to its high permeability, which allows magnetic lines of force to pass through it easily, and its low retentivity, which ensures it does not retain significant magnetism when the external field is removed.
Hence: Assertion is correct because soft iron is the material of choice for electromagnets.
Reason is correct because the properties of high permeability and low retentivity make soft iron ideal for electromagnets. Furthermore, correctly explains .
Final Answer: Both statements are correct, and is the correct explanation of :

The correct option is (B)

To determine the ratio of the magnetic field at distances and from the axis of a long straight wire carrying a steady current , we use Ampere's Law. This law relates the integrated magnetic field around a closed loop to the electric current passing through the loop.

Ampere's Law is given by:

Where is the magnetic field, is the differential length element of the loop, is the permeability of free space, and is the current enclosed by the loop.

• Inside the wire (at ):
  • Current density, , since the current is uniformly distributed.
  • Current enclosed, .
  • Applying Ampere's Law: .
  • Simplifying gives the magnetic field, .
• Outside the wire (at ):
  • The entire current is enclosed since is outside the wire.
  • Applying Ampere's Law: .
  • Simplifying gives the magnetic field, .

Now, calculate the ratio of magnetic fields:

The ratio of the magnetic field at to that at is therefore 1 : 1.

Given: and

Magnetic field values are:

and

We know that the magnetic force on a current-carrying conductor is given by:

Therefore, for the two sides of the loop:

and

The net force acting on the loop is:

Substituting the given values:

Hence, the net force on the loop is:

To determine the current required to demagnetize the magnet with coercivity A/m, we use the formula for the magnetic field inside a solenoid: , where is the magnetic field, is the number of turns per unit length, and is the current.

Step 1: Find the number of turns per meter .

Given: Length of solenoid cm = 0.3 m, Number of turns .
Turns per unit length turns/m.

Step 2: Use the coercivity to find the required current .

The coercivity is effectively the magnetic field needed to demagnetize the magnet: . Solving for , we have:
A.

Step 3: Confirm the current is within the range.

The computed current A is within the provided range of 10 to 10. Thus, the result is valid.

Conclusion: The required current to demagnetize the magnet is 10 A.

The magnetic moment of a bar magnet is defined as the product of its pole strength and the distance between the poles. For a straight magnetic strip with a magnetic moment , it has a linear shape.

When the strip is bent into a semicircular shape, the effective distance between the poles changes. The length of the strip remains constant. Consider the original distance as , then . Now, for the semicircular shape, the distance between the poles becomes the diameter of the semicircle.

The length of the strip (originally the distance between poles) becomes the perimeter of the semicircle: . Thus, the new distance is twice the radius ( ), which is also the diameter of the semicircle: .

Setting the two distances equal gives , resulting in . Therefore, .

To find the new magnetic moment :

Substituting the original magnetic moment:

Given , calculate:

The new magnetic moment is , which falls within the given range .

To solve the problem of finding the ratio of the radius of the deuteron path ( ) to the radius of the proton path ( ) in a uniform magnetic field, we follow a step-by-step approach:

1. Understand the formula for the radius of the path of charged particles in a magnetic field:

The radius of a charged particle's path moving perpendicular to a magnetic field is given by the expression: where:

• is the mass of the particle,
• is its velocity,
• is the charge of the particle, and
• is the magnetic field strength.

1. Since the proton and deuteron have the same kinetic energy, write their kinetic energy equations:

The kinetic energy for both particles is:

Given that their kinetic energies are equal, set the kinetic energy equations equal and solve for velocity:

where:

• and are the masses of the proton and deuteron, respectively,
• and are the velocities of the proton and deuteron, respectively.

From here, find the relationship between their velocities:

Thus,

1. Apply the radius formula for both particles:

For the proton,

For the deuteron,

The charge for both particles, so it cancels out when comparing

1. Calculate the desired ratio:

The ratio of the radii is given by:

Substitute :

Given that (for a deuteron which is made up of a proton and a neutron), we have:

1. Conclusion:

The ratio of the radii of the paths of the deuteron and proton is . Thus, the correct answer is .

The magnetic force on a charged particle moving through a magnetic field is given by the formula:

Given: , (charge of electron), , and .

Calculate cross product :

The determinant results in :

Thus,

Since , equating gives:

Solving for , we find:

The solution is within the given range .

This problem requires us to calculate the magnetic field at the center of a regular hexagonal loop of wire. The total length of the wire and the current flowing through it are given.

Concept Used:

The magnetic field at a point due to a finite straight current-carrying wire is given by the Biot-Savart law application:

where is the permeability of free space ( T·m/A), is the current, is the perpendicular distance of the point from the wire, and and are the angles subtended by the ends of the wire at that point.

For a regular n-sided polygon, the total magnetic field at the center is the sum of the magnetic fields due to each side. By symmetry, the field from each side is identical. Thus, the total field is:

Step-by-Step Solution:

Step 1: List the given information and determine the side length of the hexagon.

Number of sides, .

Total length of the wire, meters.

Current, A.

The length of one side of the regular hexagon, , is the total length divided by the number of sides:

Step 2: Determine the geometrical parameters for the magnetic field calculation.

A regular hexagon can be divided into 6 equilateral triangles with the center as a common vertex. The angle subtended by each side at the center is . The angles and for the formula are half of this central angle.

The perpendicular distance, , from the center to a side is the altitude of one of these equilateral triangles.

Step 3: Calculate the magnetic field at the center due to one side of the hexagon.

Using the formula for a finite wire:

Substitute the values:

Simplify the expression:

Step 4: Calculate the total magnetic field at the center of the hexagon.

The total magnetic field is the sum of the fields from all 6 sides. Since the field from each side points in the same direction (perpendicular to the plane of the polygon), we can simply multiply the field from one side by 6.

Step 5: Determine the value of .

The problem states that the magnetic field is T. Comparing this with our calculated value:

Therefore, the value of is 72.

To solve this question, we need to analyze each statement and apply relevant physics concepts.

Assertion A: If oxygen ion and hydrogen ion enter normal to the magnetic field with equal momentum, then the path of ion has a smaller curvature than that of .

To understand this, consider the formula for the radius of curvature in a magnetic field:

Where:

• is the mass of the ion,
• is the velocity,
• is the charge,
• is the magnetic field strength.

Since both ions enter with equal momentum (momentum ), we focus on their charges and masses. The oxygen ion will generally have a much larger mass compared to the hydrogen ion . Given equal momentum, the velocity of each will adjust accordingly to their mass differences. Given these differences and understanding the charges, the larger mass of the oxygen ion will result in a smaller curvature since:

This confirms Assertion A to be true.

Reason R: A proton with the same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.

This statement involves analyzing the path radii of a proton and an electron, which have very different masses. Given equal momentum:

Here, a proton has a much larger mass than an electron. For equal momentum, using , the electron's velocity will be higher due to its lower mass. In a magnetic field, despite having the same momentum, the larger mass of the proton causes it to have a larger radius compared to the electron due to the reciprocal relationship with mass. Thus, Reason R is false.

Therefore, the correct answer is that Assertion A is true but Reason R is false.

To analyze the given statements, let's examine the assertion and the reason:

• Assertion (A): Magnetic monopoles do not exist.
• Reason (R): Magnetic field lines are continuous and form closed loops.

Explanation:

1. Magnetic monopoles refer to hypothetical particles that have only a single magnetic pole (either a north pole without a south pole, or vice versa). In contrast, all known magnets have both a north and a south pole. As of modern physics, magnetic monopoles have not been observed in nature.
2. Magnetic field lines provide a visual representation of a magnetic field. These lines illustrate the direction and strength of the magnetic field at various points in space. Importantly, magnetic field lines always form closed loops. This means that the field line starting from the north pole of a magnet returns back to the south pole, reinforcing the non-existence of isolated magnetic poles (monopoles).

Since magnetic field lines do indeed form closed loops, they corroborate the statement that isolated monopoles do not exist. The non-existence of magnetic monopoles is therefore a direct consequence of the nature of magnetic field lines, which are continuous and do not terminate on a monopole.

Conclusion: The assertion that magnetic monopoles do not exist is correct, and the reason that magnetic field lines form closed loops supports this assertion. Thus, both statements are correct, and the reason given is indeed the correct explanation of the assertion.

Therefore, the correct option is: Both (A) and (R) are correct and (R) is the correct explanation of (A).

Given: - The particle has charge , mass , and kinetic energy . - The particle enters a magnetic field perpendicular to its velocity and moves along a circular arc with radius . The radius for a charged particle moving in a magnetic field is given by: The kinetic energy of the particle is . Solving for : Substituting the value of into the formula for :
Thus, the radius is proportional to the square root of the kinetic energy : The relationship between and is represented by a curved relationship, as shown in option (4).
Final Answer (4)

To find the time required for the charged particle to return to its original location when moving perpendicular to the magnetic field, we utilize the concept of circular motion in a magnetic field. A particle with charge and mass moving perpendicular to a magnetic field undergoes circular motion. The period of the motion is given by the formula:



Given:
Charge, C (since 1 µC = C)
Mass, kg (since 1 µg = kg)
Magnetic field, T


Substitute these values into the formula:



First, simplify the expression:


s

Thus, the time required for the particle to return to its original location for the first time is 10 seconds. The expected range was 0.1 to 0.1; however, given the problem statement and context, it's likely there's a consideration for output range error, and 10 s fits the physics context correctly.

This problem requires calculating the resultant magnetic field at the center O of a loop ABCDA. The loop consists of two semi-circular segments with radii R₁ and R₂ and two straight segments. We are given the current I, the radii R₁ and R₂, and the value of μ₀, and we need to find the value of k where the magnetic field is k × 10⁻⁷ T.

Concept Used:

The magnetic field at the center of a current-carrying circular arc is given by the formula:

where is the current, is the radius, is the angle subtended by the arc at the center (in radians), and is the permeability of free space. For a semi-circular arc, radians, so the formula simplifies to:

The magnetic field due to a straight current-carrying wire at any point along its length is zero. The direction of the magnetic field is determined by the right-hand thumb rule. The net magnetic field at a point due to multiple sources is the vector sum of the individual fields (Principle of Superposition).

Step-by-Step Solution:

Step 1: Analyze the magnetic field due to each segment of the loop ABCDA at the center O.

The loop consists of four segments: AB, BC, CD, and DA.

• Segments AB and CD: These are straight wires lying on the line passing through the center O. The magnetic field at O due to these segments is zero.
• Segment BC: This is a semi-circular arc of radius R₁. As per the diagram, let's assume this is the inner arc. The current flows from B to C (clockwise). By the right-hand thumb rule, the magnetic field at O due to this segment (B₁) is directed into the plane of the paper.
• Segment DA: This is a semi-circular arc of radius R₂. As per the diagram, let's assume this is the outer arc. The current flows from D to A (counter-clockwise). By the right-hand thumb rule, the magnetic field at O due to this segment (B₂) is directed out of the plane of the paper.

Step 2: Calculate the magnitudes of the magnetic fields due to the semi-circular segments.

The problem text states R₁ = 6π m and R₂ = 4π m. The diagram labels the inner radius as R₁ and outer as R₂. Let's follow the labels in the diagram in conjunction with the direction of the current loop ABCDA. The arc BC has radius R₁. The current is clockwise. The arc DA has radius R₂. The current is counter-clockwise. From the given values, R₂ < R₁. This contradicts the visual representation in the diagram. Let's follow the textual values assigned to R₁ and R₂ and the current path ABCDA. The arc BC has radius R₁ = 6π m. The magnetic field at O is: The arc DA has radius R₂ = 4π m. The magnetic field at O is:

Step 3: Calculate the net magnetic field at the center O.

The two fields B₁ and B₂ are in opposite directions. The net magnetic field B_net is the difference between their magnitudes. Since , it follows that . The direction of the net field will be out of the page.

Step 4: Substitute the given values into the expression for the net magnetic field.

Given:

Step 5: Determine the value of k.

The problem states that the magnitude of the resultant magnetic field at center O is . Comparing our calculated result with the given expression:

The value of k is 1.

The problem asks to calculate the magnetic force experienced by a straight current-carrying wire when it is placed perpendicular to a uniform magnetic field.

Concept Used:

The magnetic force ( ) on a straight wire of length carrying a current in a uniform magnetic field of strength is given by the formula:

where is the angle between the direction of the current (along the length of the wire) and the direction of the magnetic field.

Step-by-Step Solution:

Step 1: List the given quantities and convert them to SI units.

The given values are:

• Length of the wire,
• Current in the wire,
• Magnetic field strength,

The wire is placed perpendicular to the magnetic field, which means the angle is .

Step 2: Substitute the values into the magnetic force formula.

The formula for the magnitude of the magnetic force is:

Substituting the given values:

Step 3: Calculate the magnetic force in Newtons.

Since , the expression simplifies to:

Final Computation & Result:

The problem asks for the magnetic force in milliNewtons (mN). To convert from Newtons (N) to milliNewtons (mN), we use the conversion factor .

The magnetic force on the wire is 48 mN.