What is $dB/dx$ where $B$ is magnetic field and $x$ is the separation between two magnetic dipoles?

Question

I came across a question regarding the force between two magnetic dipoles $M_1$ and $M_2$ separated by a distance $x$ .

Here in this text book I am given the solution is

$$ B = \cfrac{\mu_0}{4 \pi} \cfrac{2M_1}{x^3}$$ which on differentiating with respect to distance $x$ gives

$$\frac{dB}{dx} =\cfrac{\mu_0}{4 \pi} \cfrac{6M_1}{x^4} $$ Further,

$$ F = -M_2 \frac{dB}{dx} = \cfrac{\mu_0}{4 \pi} \cfrac{6M_1 M_2}{x^4} $$

What is correct explanation of relation of '$\frac{dB}{dx}$' with force here?

Answer 1

In school we learn that energy (or work) is given by $E = -\int F \cdot dx$. Hence, we can write $dE = -F \cdot dx$. From here you see, that $F = -\frac{dE}{dx}$. All you have to do is to compare your equation with mine, $dE = M \cdot dB$, and you realise, that the magnetic dipole moment times the B-field is an energy. The term $dB/dx$ is the "gradient of the magnetic field" (first derivative).