show that in the limit that the radius of the sphere goes to zero to kept the magnetic moment constant
To show that the magnetic moment is kept constant as the radius of the sphere goes to zero, we can use mathematical reasoning.
The magnetic moment of a sphere is given by the formula:
μ = (4/3)πr³M
where μ is the magnetic moment, r is the radius of the sphere, and M is the magnetization.
In the limit as r approaches zero, we can rewrite the equation as:
lim(r→0) (4/3)πr³M = C
where C is a constant.
To evaluate this limit, we can rearrange the equation:
lim(r→0) (r³M) = C/(4/3)π
Now, as r approaches zero, we know that r³ also approaches zero, regardless of the value of M. Therefore, the product r³M approaches zero.
So, in the limit as the radius of the sphere goes to zero, the magnetic moment is kept constant and equal to C/(4/3)π.
Note: This explanation assumes that the magnetization M remains finite and does not change as the radius approaches zero.