Net magnetic field due to short barmagnets

[zorro]

Homework Statement

Two short magnets of equal dipole moments M are fastened perpendicularly at their centres as shown. The magnetic field (magnitude) at a point P, distant d from the centre on the bisector of the right angle is ?

[URL]http://203.196.176.41/VLEBT_RootRepository/Resources/8f3a3668-e3a2-45c9-beae-68d195a290f4.gif[/URL]

The Attempt at a Solution

I am confused on finding out the magnitude of the net magnetic field.
The magnitude of the magnetic field due to each magnet (I don't know their direction) is μM/(4√2πd²)

 

[rl.bhat]

The combination of two short magnets a shown in the problem has an effective dipole moment equal to sqrt(2)*M along the bisector of the right angle.

 

[zorro]

That is a great approach!
So the resultant dipole moment is √2M and the resultant field is μ2√2M/4πd²

 

[rl.bhat]

That is a great approach!
So the resultant dipole moment is √2M and the resultant field is μ2√2M/4πd²

No. For short magnet, the field is inversely proportional to d³

 

[zorro]

I copy-pasted the denominator from previous expression and forgot to change the square to cube. It should be μ2√2M/4πd³

 

[hikaru1221]

The combination of two short magnets a shown in the problem has an effective dipole moment equal to sqrt(2)*M along the bisector of the right angle.

Note for Abdul: This approach is only correct for this problem. It's not true in general, not even in another problem with another pair of magnets of some different configurations.

 

[zorro]

not even in another problem with another pair of magnets of some different configurations.

Like?

 

[hikaru1221]

2 magnets positioned at some distance from each other, provided that the point of interest P is not equidistant from the magnets and the medium is homogenous. You want to guess the reason?

 

[zorro]

May be this approach is fine only when the point lies on the line of resultant dipole moment?

 

[hikaru1221]

Nope. Think a bit hard: The essence of the approach is to combine 2 moments. If the approach is not applicable, the resultant moment should not "exist" or have some sort of meaning. Therefore, talking about the resultant moment (or "when the point lies on the line of resultant dipole moment" in particular) doesn't make much sense.

So that's a hint: it's not about the resultant moment. It should be about why the resultant moment should not "exist" here.

A further hint: B-field by M depends on M and the distance d from M to the point of interest P.

P.S.: M denotes vector M. I'm not sure what's wrong with LaTex