The magnetic moment of a thin bar magnet is 'M'. If it is bent into a semicircular form, its new magnetic moment will be:
2M/π
Understanding the Magnetic Moment of a Bar Magnet
The magnetic moment of a bar magnet is a measure of its strength and orientation. For a thin bar magnet, the magnetic moment (M) is defined as the product of the strength of each pole (pole strength, \(q_m\)) and the distance between the poles (the length of the magnet, L).
Mathematically, the initial magnetic moment is:
\( M = q_m \times L \)
Analyzing the Bar Magnet Bent into a Semicircle
When the thin bar magnet is bent into a semicircular form, the pole strength (\(q_m\)) at the ends of the magnet remains the same. However, the distance between the poles changes significantly.
- Initially, the distance between the poles was the straight length 'L'.
- After bending into a semicircle, the poles are located at the two ends of the semicircle. The shortest distance between these two points is the diameter of the circle from which the semicircle is formed.
Relating Original Length to Semicircle Dimensions
The original length 'L' of the bar magnet now forms the arc length of the semicircle. If 'R' is the radius of this semicircle, the arc length of a semicircle is given by \(\pi R\). Therefore, we have:
\( L = \pi R \)
From this relationship, we can find the radius 'R' in terms of the original length 'L':
\( R = \frac{L}{\pi} \)
Calculating the New Distance Between Poles
The new distance between the poles is the diameter of the semicircle, which is \(2R\). Substituting the value of R we just found:
New distance between poles = \( 2R = 2 \times \left(\frac{L}{\pi}\right) = \frac{2L}{\pi} \)
Calculating the New Magnetic Moment
The new magnetic moment (\(M'\)) of the bent magnet is the product of the pole strength (\(q_m\)) and the new distance between the poles:
\( M' = q_m \times \left(\frac{2L}{\pi}\right) \)
We know that the original magnetic moment was \( M = q_m \times L \). From this, we can express the pole strength \(q_m\) as \(q_m = \frac{M}{L}\).
Now, substitute this expression for \(q_m\) into the equation for \(M'\):
\( M' = \left(\frac{M}{L}\right) \times \left(\frac{2L}{\pi}\right) \)
The 'L' in the numerator and denominator cancels out:
\( M' = M \times \frac{2}{\pi} \)
\( M' = \frac{2M}{\pi} \)
Conclusion on New Magnetic Moment
When a thin bar magnet of magnetic moment 'M' is bent into a semicircular form, its new magnetic moment becomes \( \frac{2M}{\pi} \).
| Property | Original Bar Magnet | Semicircular Magnet |
|---|---|---|
| Magnetic Moment | \( M = q_m L \) | \( M' = q_m (\frac{2L}{\pi}) \) |
| Pole Strength | \( q_m \) | \( q_m \) (Remains same) |
| Distance between Poles | \( L \) | \( \frac{2L}{\pi} \) (Diameter) |
Revision Table: Key Concepts for Magnetic Moment
| Term | Definition/Formula | Units (SI) |
|---|---|---|
| Magnetic Moment (\(M\)) | \( q_m \times L \) (for bar magnet) \( I \times A \) (for current loop) |
Ampere-meter<sup>2</sup> (A m<sup>2</sup>) |
| Pole Strength (\(q_m\)) | A measure of the strength of a magnetic pole. | Ampere-meter (A m) |
| Magnetic Dipole | A pair of equal and opposite magnetic poles separated by a distance. A bar magnet acts as a magnetic dipole. | - |
Additional Information: Magnetic Dipole Moment
The magnetic moment is often referred to as the magnetic dipole moment. It is a vector quantity, pointing from the south pole to the north pole of the magnet. In the case of the bent magnet, the poles are at the ends of the semicircle, and the new magnetic moment vector points along the diameter connecting these ends.
The calculation above relies on the assumption that the pole strength is concentrated at the ends of the bar magnet, which is a common model for thin magnets. Bending the magnet changes the spatial arrangement of the poles, thus altering the effective dipole length (the distance between the poles) and consequently the magnetic moment.
Important Questions from Current Electricity
The current through a 4/3 Ω external resistance connected to a parallel combination of two cells of 2 V and 1 V emf and internal resistances of 1 Ω and 2 Ω respectively is:
A metallic wire of uniform area of cross-section has a resistance R, resistivity ρ, and power rating P at V volts. The wire is uniformly stretched to reduce the radius to half the original radius. The values of resistance, resistivity, and power rating at V volts are now denoted by R', ρ', and P' respectively. The corresponding values are correctly related as:
A cell of emf 1.1 V and internal resistance 0.5 Ω is connected to a wire of resistance 0.5 Ω. Another cell of the same emf is now connected in series with the intention of increasing the current, but the current in the wire remains the same. The internal resistance of the second cell is:
P, Q, R, and S are four wires of resistances 3 Ω, 3 Ω, 3 Ω, and 4 Ω, respectively. They are connected to form the four arms of a Wheatstone bridge circuit. The resistance with which S must be shunted in order that the bridge may be balanced is:
The current flowing through the two bulbs marked as 60W, 240V each when connected in series with a 240V source is:
