Magnetism And Matter

The magnetic field at a point on the axis of a bar magnet is given by the formula: where:
- is the permeability of free space,
- is the magnetic moment of the bar magnet,
- is the distance from the center of the magnet.
Thus, the magnetic field is directly proportional to the magnetic moment and inversely proportional to the cube of the distance from the magnet.

Step 1: Relate the components of the Earth's magnetic field.
Let be the total intensity of the Earth's magnetic field. Let be the horizontal component and be the vertical component. The total field is the vector sum of its components: The angle of dip, , is the angle between the total field and the horizontal component, defined by:

Step 2: Use the given condition to find the ratio of components.
We are given that the Earth's magnetic field is twice its vertical component: Substitute this into the total field equation:

Step 3: Find the relationship between and .
Take the square root of the equation from Step 2:

Step 4: Calculate the required ratio.
The question asks for the ratio of the horizontal component to the total magnetic field, . From the given information, we have and . Substitute in the first relation: Rearrange to find the ratio : The ratio is .

There are two related quantities inside a solenoid: magnetic field ( ) and magnetic intensity ( ).
Magnetic Field: , where is the permeability of the core material.
Magnetic Intensity: .
The magnetic intensity depends only on the number of turns per unit length ( ) and the current ( ), and is independent of the core material.
We are given:
Number of turns per metre, turns/m.
Current, A.
The relative permeability is extra information, not needed for calculating .
Now, we calculate the magnetic intensity :
.
A/m or Am .
This can be written in scientific notation as Am .

Step 1: Understanding the Concept:
This problem involves the superposition of magnetic fields. We need to find the magnetic field produced by each of the two short bar magnets at point P and then find their vector sum to get the resultant field.
Step 2: Key Formula or Approach:
For a short bar magnet of magnetic moment , at a distance from its center:
- Magnetic field on its axis: .
- Magnetic field on its equatorial line: .
The resultant field from two perpendicular vectors and is .
Step 3: Detailed Explanation:
Let the first magnet (moment M) be placed along the x-axis with its center at the origin. Point P is on its axis at a distance , so P = (r, 0).
The magnetic field at P due to the first magnet is , directed along the x-axis.
We are given that this field has a magnitude . So, .
The second magnet (moment 2M) is placed with its center at the origin and its axis perpendicular to the first magnet. Let's say its axis is along the y-axis.
For this second magnet, the point P = (r, 0) lies on its equatorial line.
The magnetic field at P due to the second magnet is , directed along the y-axis (or -y, but perpendicular to ).
By comparing the expressions for and , we can see that their magnitudes are equal:
The total magnetic field at P, , is the vector sum of and . Since is along the x-axis and is along the y-axis, they are perpendicular.
The magnitude of the resultant field is:
Step 4: Final Answer:
The resultant magnetic field at point P is . Therefore, option (C) is correct.

1. Magnetic moment of a bar magnet: .
2. Volume , where = area of cross-section, = length.
3. Given: Am , Am , cm m .
4. Length: m = 9 cm

• For a strong electromagnet, the core should easily magnetize under applied current; this requires high magnetic permeability.
• The core should also demagnetize quickly when the current is turned off; this requires low retentivity to avoid residual magnetism.
• Materials like soft iron satisfy both conditions.
• Hence correct choice: High permeability and low retentivity.