Why doesnt the magnetic field inside a solenoid depend on its radius whereas the magnetic field inside a circular loop or a coil depends on its radius?Please explain using simple language keeping in mind that I'm a 10th standard student.

# [Physics] Magnetic field of a solenoid vs a circular loop

applied-physicselectromagnetism

#### Related Solutions

This depends on exactly what you mean by non-uniform, or (equivalently) on how big the loop is. In particular, the important criterion is whether the field changes appreciably over distances that are about the same size as the loop.

If the field changes throughout space, but the loop is small enough that the field doesn't change much from point to point on the loop, then the uniform-field formula $\boldsymbol\tau=\boldsymbol\mu\times\mathbf B(\mathbf r)$ still applies. In essence, the field is

*locally*uniform, though the direction and magnitude it's uniform on can change from place to place.If the loop is big enough that the field changes appreciably over its span then there's nothing for it but to integrate the local torque on each bit of circuit and add them up, which gives you $$\boldsymbol\tau =\oint_\mathcal{C}\mathbf r\times\mathbf F(\mathbf r)\:\mathrm d l =\oint_\mathcal{C}\mathbf r\times(\hat{\mathbf t}I\times\mathbf B(\mathbf r))\:\mathrm d l =\oint_\mathcal{C}\mathbf r\times(I\mathrm d \mathbf l\times\mathbf B(\mathbf r)). $$ There really isn't much you can do to simplify it beyond that without special assumptions. The integral is a line integral, of exactly the same sort you use to calculate the magnetic dipole moment $\boldsymbol \mu$ itself.

The following is a bit more technical and uses a standard amount of vector calculus as used in undergraduate electromagnetism courses.

If the field were constant, then you can manipulate it more freely to get
\begin{align}
\boldsymbol\tau
=\oint_\mathcal{C}\mathbf r\times(I\mathrm d \mathbf l\times\mathbf B)
=I\oint_\mathcal{C}\left[(\mathbf B·\mathbf r)\mathrm d\mathbf l-(\mathbf r·\mathrm d\mathbf l)\mathbf B\right],
\end{align}
using a vector triple product. Here the second integral vanishes, because Stokes' theorem implies that
$$
\oint_\mathcal{C}\mathbf r·\mathrm d\mathbf l=\int_\mathcal{S}(\nabla\times\mathbf r)·\mathrm d\mathbf S=0.
$$
You're left with the first integral, $\boldsymbol\tau=I\oint_\mathcal{C}(\mathbf B·\mathbf r)\mathrm d\mathbf l$, which in component notation reads $\tau_i=IB_j\oint_\mathcal{C} x_j \mathrm dx_i$ (I use Einstein summations throughout). As it turns out, the indices on that last integral are antisymmetric, i.e.
$$\oint_\mathcal{C} x_j \mathrm dx_i=-\oint x_i \mathrm dx_j,$$
which you can again prove via Stokes' theorem since
\begin{align}
\oint_\mathcal{C}\left[ x_j \mathrm dx_i+x_i\mathrm dx_j\right]
=\oint_\mathcal{C}\left[ x_j (\nabla x_i)·\mathrm d\mathbf l+x_i(\nabla x_j)·\mathrm d\mathbf l\right]
=\oint_\mathcal{C}\nabla(x_ix_j)·\mathrm d\mathbf l
=\int_\mathcal{S}\nabla\times\nabla(x_ix_j)·\mathrm d\mathbf S
=0.
\end{align}
What this means is that you can "fold" the old integral into two antisymmetric parts, as
$$\tau_i=IB_j\oint_\mathcal{C} \frac{x_j \mathrm dx_i-x_i \mathrm dx_j}{2},$$
and the antisymmetrized integral is now exactly the magnetic dipole moment
$$
\boldsymbol\mu=\frac I2\oint_\mathcal C\mathbf r\times\mathrm d\mathbf l
$$
which in component notation reads
$$
\mu_k=I\oint_\mathcal C\varepsilon_{kij}x_i\mathrm dx_j
,\quad\text{or in other words}\quad
\mu_k\varepsilon_{kij}=I\oint_\mathcal C \left[x_i\mathrm dx_j-x_j\mathrm dx_i\right].
$$
Back to the torque this means
$$\tau_i=\frac12 B_j\varepsilon_{kji}\mu_k,$$
or in vector notation
$$
\boldsymbol\tau=\boldsymbol\mu\times\mathbf B.
$$
So what is it I've done? All of this work has been to take an integral that had the magnetic field inside it, and factorize it into a system-dependent part and a field dependent part:
$$
\boldsymbol\tau
=\oint_\mathcal{C}\mathbf r\times(I\mathrm d \mathbf l\times\mathbf B)
=\left(\frac I2\oint_\mathcal C\mathbf r\times\mathrm d\mathbf l\right)\times\mathbf B.
$$
That's really what the magnetic dipole moment $\boldsymbol\mu$ really *is*.

OK, sorry, that got out of hand, but I'll get back on topic now. What does this have to do with *non*uniform magnetic fields? Well, if your field varies very slowly with respect to the dimensions of the loop, you can suppose that $\mathbf B$ is constant in your integral, and you get the calculation above. The next thing you might try is suppose that $\mathbf B$ is *almost* constant throughout the extent of the loop, but that you do need to consider the first-order term of its Taylor series. Thus, you could suppose that
$$
\mathbf B(\mathbf r)=\mathbf B(\mathbf r_0)+(\mathbf r·\nabla)\mathbf B(\mathbf r_0),
$$
or more clearly in component notation
$$
B_i(\mathbf r)=B_i(\mathbf r_0)+x_k\frac{\partial B_i}{\partial x_k}(\mathbf r_0).
$$

You can then do the same game I've done above with the second term, with the added complication that you have an extra factor of $x_k$ in your integral. The result will be something which depends on the first-order derivative of the field, $\frac{\partial B_i}{\partial x_k}(\mathbf r_0)$, and if you're clever you can factorize out the dependence on the loop into a single factor. This factor will in general be (i) a matrix, or in fancy-speak a tensor, (ii) the integral of a homogeneous second-degree polynomial over the loop, and (iii) it is called the **quadrupole moment** of the loop.

In general, such calculations are long and messy, but if you're feeling brave I encourage you to have a go at deriving an expression for the quadrupole moment and seeing how simple of an expression you can get for the moment itself and its interaction with the magnetic field's gradient to get the torque. As a start, though, here's one expression for the quadrupolar part of the torque, which I encourage you to derive $$ \tau_i=\frac{\partial B_m}{\partial x_k}·I\oint_\mathcal{C}\left[ x_kx_m\mathrm d x_i-\delta_{im}x_k x_j\mathrm dx_j \right]. $$

Of course, there's only so much you can do with only first-order derivatives. If your field varies slightly faster than that - or if your circuit is somewhat bigger - then the next thing you can try is a second-order Taylor expansion of the field. This gives you a third term which depends on the second derivatives of the magnetic field and on a system-dependent term which is (i) an unwieldy object with three different indices, called a rank-3 tensor, (ii) the integral of a homogeneous third-degree polynomial over the loop, and (iii) is called the **octupole moment** of the loop. And after that, you can go to even higher orders, and on and on it goes until you feel your calculation is accurate enough, or you give up through exhaustion.

I can tell you, though - none of the formulas there are pretty. That's why they're hard to find online.

Parallel currents attract. The solenoid, with current in the direction shown, has current parallel to that in the loop, and the loop will be attracted to the solenoid. A uniform B field, however, would put no net force on such a loop; any force on one section of the loop would be cancelled by equal and opposite force on another section of the loop (the opposite side of the loop carrying the same current in the opposite direction).

The radial (radiating as if it were coming from a point in the center) B field component is not uniform (it's strong in the center, and points in various directions). That part of the B field puts a net force on the current-carrying ring.

## Best Answer

Circular loopAs you stated, the magnetic field inside a circular loop depends as the inverse of it's square radius. This a direct result from the calculation of the magnetic field, which can be obtained using the Biot-Savart formula. Don't worry if you haven't studied it yet, because what is important is that the bigger the circular coil, the smaller the magnetic field becomes for a given current. This can be better visualized with the following image from Hyperphysics:

SolenoidYou've been taught that the magnetic field of a solenoid, on the other hand, doesn't depend on its radius. However, this is just an idealization that applies for a perfect solenoid, that is, a solenoid which has a small radius and is infinitely long. This allows one to consider all the lines of the magnetic field as parallel all along the inside of the solenoid, which traduces in the magnetic field being constant everywheres inside the solenoid. This image from Quora will help you visualize it:

I think the best analogy for the magnetic field in a solenoid would be a tube (with a constant diameter) through which water flows in it. If the flux is perfect, you will measure the same velocity of water no matter where you put yourself inside the tube. In this case, the tube is your solenoid, and the velocity of water corresponds to the magnetic field.

However, in real life solenoid the magnetic field does indeed vary with the radius of the solenoid, and you could measure certain variations if you deviate from the axis of symmetry of the solenoid, like this image taken from Wikipedia:

Each blue line you see represents the value and direction of the magnetic field inside and outside of the solenoid. Since here the diameter of the solenoid is bigger than its length, you can see that as we approach the border the magnetic field does change. Therefore, the best solenoids are those who have a small diameter in comparison with their length.