Magnetic Effects Of Electric Current

Step 1: A current loop in a magnetic field experiences a force that tends to maximize the magnetic flux passing through it.
Step 2: For a given perimeter, a circle has the maximum area. Thus, the flexible wire expands into a circular shape.
Step 3: To maximize flux ( ), the plane of the loop aligns perpendicular to the field ( ).

Step 1: Understanding the Concept:
The magnetic field at the center of a single circular loop of radius carrying current is given by .
For a spiral coil, we treat it as a collection of infinitesimal concentric loops and integrate their contributions.
Step 2: Key Formula or Approach:
Number of turns per unit radial thickness is .
For a small radial element at distance from the center, the number of turns is .
Step 3: Detailed Explanation:
The infinitesimal magnetic field at the center due to the turns in width is:

Total magnetic field is obtained by integrating from to :




Step 4: Final Answer:
The magnetic field at the center is .

Step 1: Understanding the Question:
Two ions with different masses and charges enter a uniform magnetic field with the same kinetic energy. We need to compare their deflection. Deflection is inversely related to the radius of the circular path they follow. A smaller radius means a larger deflection.
Step 2: Key Formula or Approach:
When a charged particle moves perpendicular to a magnetic field, it follows a circular path. The radius of this path is given by: where is mass, is velocity, is charge, and is the magnetic field strength.
The kinetic energy is . We can express momentum in terms of kinetic energy: .
Substituting this into the radius formula: Step 3: Detailed Explanation:
Let the lighter ion be ion 1 and the heavier ion be ion 2.
Given data:
Mass of lighter ion, amu
Charge of lighter ion,
Mass of heavier ion, amu
Charge of heavier ion,
Kinetic energy is the same for both: .
The magnetic field is also the same: .
Now we calculate the ratio of their radii. Since and are constant, the radius is proportional to .
Radius of the lighter ion's path: Radius of the heavier ion's path: Comparing the radii, we find that .
Since the radius of the path of the lighter ion ( ) is smaller than that of the heavier ion ( ), the lighter ion follows a more tightly curved path.
Step 4: Final Answer:
A more curved path means a greater deflection from the original direction of motion. Therefore, the lighter ion will be deflected more than the heavier ion.

Step 1: Understanding the Question:
A wire of fixed length is first used to make a triangular coil and then a square coil, both with side length 'a'. We need to find the ratio of their magnetic dipole moments.
Step 2: Key Formula or Approach:
The magnetic dipole moment ( ) of a current loop is given by , where: - is the number of turns in the coil. - is the current flowing through the coil. - is the area of the loop. The number of turns is the total length of the wire divided by the perimeter of one loop. The current is the same for both cases since the wire (with resistance R) is connected to the same voltage source . Thus, .
Step 3: Detailed Explanation:
Total length of the wire, .
Case 1: Equilateral Triangle Coil
Side of the triangle = . Perimeter of one turn = . Number of turns, . Area of an equilateral triangle, . Magnetic moment of the triangular coil: Case 2: Square Coil
Side of the square = . Perimeter of one turn = . Number of turns, . Area of the square, . Magnetic moment of the square coil: Ratio of Magnetic Moments
We need to find the ratio . The question states this ratio is , which means . By comparison, we get .
Step 4: Final Answer:
The value of y is 3.

Step 1: Understanding the Concept:
The magnetic field at the centroid of a current-carrying polygon is the sum of the fields produced by each individual side. For an equilateral triangle, all three sides contribute equally and in the same direction.
Step 2: Key Formula or Approach: 1. Field due to a finite straight wire: .
2. For equilateral triangle side , distance to centroid and .
Step 3: Detailed Explanation: Given , . Field due to one side: Total field at centroid . Direction: Since the current is clockwise, by the right-hand thumb rule, the magnetic field at the center points into (inside) the plane of the triangle.
Step 4: Final Answer:
The magnetic field is T, inside the plane.

Step 1: . So .
Step 2: .
Step 3: Take cube root of both sides: .
Step 4: Square both sides: .
Step 5: .
Step 6: m.

Step 1: The magnetic field inside a solenoid is . Since they are identical, .
Step 2: Solenoid A is in series with the parallel bank of B, C, and D. If current enters A, it splits equally into for B, C, and D (since they are identical).
Step 3: . .