Magnetic Effects Of Electric Current

Step 1: Magnetic field due to current-carrying wire.
The magnetic field at a point due to a current in a long straight wire is given by: where is the permeability of free space and is the distance from the wire.
Step 2: Using the given data.
When both currents are in the same direction, the magnetic fields due to both wires add up, and at the point midway between them, the total field is: Given . When is reversed, the magnetic fields due to the two currents subtract: Given . Step 3: Solving for the ratio.
From these two equations, we can solve for the ratio and find it to be 2.
Step 4: Conclusion.
Thus, the ratio is 2, corresponding to option (2).

Step 1: Understanding the force on a charged particle in a magnetic field.
The force on a charged particle moving with velocity in a magnetic field is given by the Lorentz force law: where is the angle between the velocity vector and the magnetic field vector.
Step 2: Applying the condition in this case.
In this case, the charged particle enters the solenoid and moves along the axis of the solenoid. The magnetic field inside a solenoid is along its axis, and the velocity of the particle is also along the axis. Therefore, the angle , and since , the force .
Step 3: Conclusion.
Thus, the force acting on the charged particle is zero, corresponding to option (A).

Step 1: Magnetic moment of a coil.
The magnetic moment of a circular coil is given by: where is the number of turns, is the current, and is the area of the coil. The area of the coil is , where is the radius of the coil.
Step 2: Proportionality of magnetic moment.
Thus, the magnetic moment is proportional to , corresponding to option (C).
Step 3: Conclusion.
The magnetic moment is proportional to , corresponding to option (C).

Step 1: Write dimensional formulae.
Force:
Density:

Step 2: Write the given relation in dimensional form.

Step 3: Substitute dimensions of force and density.

Step 4: Solve for dimensions of .

Step 5: Conclusion.
The dimensions of are .

Step 1: Magnetic field due to current in ring A.
For a circular loop of radius carrying current , the magnetic field at its center is given by:

Step 2: Magnetic flux through ring B.
Since , the magnetic field over ring B can be assumed uniform.
Area of ring B is .

Step 3: Mutual inductance calculation.
Mutual inductance is defined as:

Step 4: Conclusion.
The correct expression for mutual inductance is option (C).

Step 1: Recall the relation between current gains.
For a transistor, the relation between the common-base current gain and the common-emitter current gain is:

Step 2: Find the reciprocal of .

Step 3: Substitute in the given expression.

Step 4: Simplify the expression.

Step 5: Conclusion.
Hence, the correct value of the given expression is .

Step 1: Magnetic Moment Formula.
The magnetic moment for a current loop is given by: where is the current and is the area of the loop. For a semi-circular loop, the area is . The current is given by: where is the time period for one complete revolution. Thus, the magnetic moment becomes: Step 2: Final Answer.
Thus, the magnetic moment is .

Step 1: Understanding the behavior of magnetic susceptibility.
Magnetic susceptibility ( ) for a paramagnetic substance typically increases as the magnetic field ( ) is applied and reaches a saturation value. At higher field strengths, the susceptibility tends to level off, showing a near-constant behavior.
Step 2: Analyzing the options.
(A) Graph (A): This graph shows a decreasing magnetic susceptibility as the magnetic field increases, which is incorrect for a paramagnetic substance.
(B) Graph (B): Correct — This graph shows the expected behavior of a paramagnetic substance where the susceptibility increases and then levels off at higher magnetic fields.
(C) Graph (C): This graph shows a saturated behavior but at an inappropriate initial stage. It does not represent the typical trend for paramagnetic substances.
(D) Graph (D): This graph shows an oscillating trend, which is incorrect for paramagnetic substances.
Step 3: Conclusion.
The correct answer is (B), as it correctly depicts the variation of magnetic susceptibility with the applied magnetic field for a paramagnetic substance.

Step 1: Understanding the Force on the Wire.
The force on a wire carrying current due to the magnetic field from another wire is given by: where is the distance between the wires. Since the third wire is placed midway between the two wires and anti-parallel to one of the wires, the force on it will be directed towards the wire carrying the larger current (i.e., towards the wire carrying current ).
Step 2: Final Answer.
Thus, the force on the third wire will be towards the wire carrying current .

Step 1: Expression for magnetic force.
Magnetic force on a charged particle is given by
Step 2: Direction of velocity and magnetic field.
When the velocity vector is parallel to the magnetic field , the angle between them is .
Step 3: Evaluating the force.

Step 4: Conclusion.
Hence, the magnetic force acting on the particle is zero.

Step 1: Use the formula for the force between two current-carrying wires.
The force between two parallel wires carrying currents and separated by a distance is given by Ampère's law: where is the permeability of free space. Step 2: Calculate the effect of changing distance and current.
The force is directly proportional to the product of the currents and inversely proportional to the distance between the wires. If the distance is doubled and the current is reduced to one-third, the new force is: Step 3: Conclusion.
Thus, the force exerted between the wires is , which corresponds to option (C).

Step 1: Magnetic moment of a loop.
The magnetic moment of a current-carrying loop is given by: where is the current passing through the loop, and is the area of the loop. Step 2: Calculate the area of both loops.
For a square loop with side , the area is: For a circular loop with radius , the area is: Step 3: Relationship between the lengths of the loops.
The length of the square loop is , and the length of the circular loop is . Since the total length of the wire is the same for both loops, we have: Step 4: Calculate the ratio of magnetic moments.
The magnetic moment of the square loop is: The magnetic moment of the circular loop is: Thus, the ratio of magnetic moments is: Step 5: Conclusion.
Thus, the ratio of the magnetic moment of the square loop to that of the circular loop is , which corresponds to option (C).