Ampere's Law vs Biot-Savart Law: Which to Use

The choice, in one line

Every magnetic field in this course comes from a current, and these are the two routes from the current to the field. Neither is more correct than the other. The difference is entirely about what work you have to do.

The [Biot-Savart law](/glossary/biot-savart-law) is a recipe for adding. Every short segment of current-carrying wire makes its own small magnetic field, and you add up all of those contributions along the whole wire. It works on any current path whatsoever. The cost is that each contribution is a cross product, which is to say a vector with a direction that changes as you move along the wire, so you resolve into components before you integrate.

[Ampere's law](/glossary/amperes-law) is a recipe for cancelling. You draw a closed loop, and the law tells you the circulation of $\vec{B}$ around it. If the loop can be chosen so that the field magnitude is constant along it and always parallel to it, the integral collapses to $B$ times a length and you solve for $B$ in one line. The cost is that such a loop exists only for a short list of current geometries.

So the decision procedure is:

1. Long straight wire, long solenoid, or a slab or cylinder carrying a current density? Ampere.
2. Anything else, including a single loop, an arc, or a finite segment? Biot-Savart.

If that pattern sounds familiar, it should. It is exactly the relationship Gauss's law has to Coulomb's law, moved from the electric half of the course to the magnetic half. Coulomb is to Gauss as Biot-Savart is to Ampere: an always-applicable integrating law paired with a symmetry shortcut. Recognising that once means you learn one decision procedure instead of two, and it is the reason Coulomb's law vs Gauss's law is worth reading alongside this page even though it is about a different field.

Side by side

The row that decides a problem is what you must know. Biot-Savart demands every detail of the wire's shape, because each piece contributes differently. Ampere demands only a number, the enclosed current, and hands you the field without your knowing how the current is distributed inside the loop, as long as the distribution is symmetric.

The row people skip is the $1/r^2$ against $1/r$. Each element of current produces an inverse-square field, exactly like a point charge. The field of an infinite wire falls off as $1/r$ instead, and that is not a contradiction: it is what you get after adding infinitely many inverse-square contributions from a line that extends forever. The same thing happens on the electric side, where a point charge gives $1/r^2$ and an infinite line of charge gives $1/r$.

What the CED lets each one do

Both topics carry boundary statements, and reading them together tells you which problems are which before you start.

Biot-Savart, from the Topic 12.3 boundary statement: AP Physics C: Electricity and Magnetism only expects students to perform quantitative analysis of certain cases of current-carrying conductors using the Biot-Savart law, such as at a location along the perpendicular bisector of a straight conductor, at a location along the central axis of a circular loop, or at the centre of a segment of a circular loop.

Ampere, from the Topic 12.4 boundary statement: AP Physics C: Electricity and Magnetism only expects quantitative application of Ampere's law limited to situations involving symmetrical magnetic fields, and it names the shapes: long straight wires, long solenoids carrying currents, and conductive slabs or cylindrical conductors carrying a current density.

Put the two lists side by side and three things stand out.

• The two lists barely overlap. A circular loop and an arc appear only on the Biot-Savart list. A solenoid and a slab appear only on the Ampere list. The exam is not asking you to choose between them very often; it is asking you to recognise which one the geometry already chose.
• The straight wire is on both, in different guises: the perpendicular bisector of a finite straight conductor for Biot-Savart, the long straight wire for Ampere. That overlap is the standard exercise for showing that the two routes agree, and worked example one does it.
• Every Biot-Savart case on that list is one where the cross product is easy. Along the perpendicular bisector, at the centre of a loop, at the centre of an arc: in all three, either $d\vec{\ell}$ is perpendicular to $\hat{r}$ everywhere or the distance $r$ is constant, or both. The CED is not asking for hard integrals; it is asking for the ones where symmetry does most of the cross product for you.

That last point has a check you can run yourself. The C: E&M sheet's calculus table prints five integrals: $\int x^n dx$, $\int e^{ax} dx$, $\int \frac{dx}{x+a}$, $\int \cos(ax) dx$ and $\int \sin(ax) dx$. The integral you would need for a general straight segment, $\int \frac{dx}{(x^2+a^2)^{3/2}}$, is not among them. The calculus formulas reference lists the printed set. The exam sets Biot-Savart problems whose integrals you can do without it.

The overlap: a long straight wire, both ways

This is the one geometry where you genuinely have a choice, and comparing the two routes on it is the fastest way to feel the difference in cost.

Ampere. The field circles the wire and has the same magnitude at every point of a circle of radius $r$ centred on it. So $\oint \vec{B} \cdot d\vec{\ell} = B(2\pi r) = \mu_0 I$, and $B = \frac{\mu_0 I}{2\pi r}$. Three lines, one of which is a symmetry argument. The CED prints this as a derived equation under essential knowledge 12.4.A.1.i.

Biot-Savart. Set up coordinates along the wire, write each element's contribution as $dB = \frac{\mu_0}{4\pi}\frac{I\,dx\sin\theta}{r^2}$, notice that every element's field points the same way at the field point so the vector sum becomes a scalar one, express $\sin\theta$ and $r$ in terms of $x$ and the perpendicular distance $a$, and integrate from minus infinity to infinity. You arrive at $B = \frac{\mu_0 I}{2\pi a}$, the same answer, after roughly a page.

Same result, wildly different effort, which is the whole argument for learning Ampere's law at all. And the reason the effort differs is worth naming: Biot-Savart never used the symmetry. It rebuilt the answer element by element, and the symmetry only showed up at the end as an integral that happened to be doable. Ampere's law uses the symmetry at the start, as an argument about what the field can possibly look like, and never integrates anything.

One caution. Because the wire example works both ways, it is tempting to conclude that Ampere's law is always available and just harder to see. It is not. The next section is the counterexample.

The case that separates them: the centre of a circular loop

Bend the wire into a circle of radius $R$ and ask for the field at the centre. Now try Ampere's law.

What closed path has $\vec{B}$ constant along it and parallel to it? Not a circle in the plane of the loop, because the field there is perpendicular to the plane, not tangent to any circle drawn in it. Not a circle around the wire itself, because that circle passes through regions where the other parts of the loop contribute differently. There is no usable Amperian loop, so Ampere's law is still true and gives you nothing.

Biot-Savart, on the other hand, finishes almost instantly, and for a specific reason: at the centre of the loop, every element $d\vec{\ell}$ is perpendicular to its $\hat{r}$, so $\lvert d\vec{\ell} \times \hat{r} \rvert = d\ell$ with no sine to carry, and every element is the same distance $R$ away. The integral reduces to $\int d\ell = 2\pi R$, and you get

which the CED prints as a derived equation under essential knowledge 12.3.A.3. Worked example two does the arithmetic.

The same argument, run over a fraction of a circle instead of a whole one, handles the arc case on the CED's list: a semicircular arc contributes half as much, a quarter arc a quarter as much, because the only thing the integral counted was arc length. That gives you a family of answers with no new work, and no version of Ampere's law reaches any of them.

So the honest summary is not that Ampere's law is a faster Biot-Savart. It is that Ampere's law solves a narrow set of problems almost for free and is unavailable outside it, while Biot-Savart solves everything at a price.

What the C: E&M equation sheet prints

Everything in this comparison sits on one page, the Electricity and Magnetism sheet, which is one of the four sheets in the C: E&M booklet alongside the constants page, the mechanics page, and the geometry, vectors, calculus and identities page.

From the magnetism block:

• $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I(d\vec{\ell} \times \hat{r})}{r^2}$, the Biot-Savart law, in differential form, for one element
• $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\mathrm{enc}}$, Ampere's law
• $B_{\mathrm{sol}} = \mu_0 n I$, the solenoid field, where $n$ is turns per unit length
• $\vec{F}_B = \int I(d\vec{\ell} \times \vec{B})$, the force on a current-carrying wire, which is a different integral and is easy to confuse with Biot-Savart because it has the same shape
• $\oint \vec{B} \cdot d\vec{A} = 0$, Gauss's law for magnetism

The constants page prints $\mu_0 = 4\pi \times 10^{-7} \ \mathrm{(T \cdot m)/A}$. In practice, carry $\frac{\mu_0}{4\pi} = 1.0 \times 10^{-7}$ and $\frac{\mu_0}{2\pi} = 2.0 \times 10^{-7}$ in your head; both appear constantly and both are exact to the sheet's precision.

Three observations that repay a second look.

Biot-Savart is printed as $d\vec{B}$, not $\vec{B}$. The sheet is handing you an integrand. Nothing on the sheet performs the integral for you, which is the sheet's way of saying that the setup is the graded work.

$B_{\mathrm{sol}} = \mu_0 n I$ is printed as a finished result. The CED reaches it as a derived equation under essential knowledge 12.4.A.1.iii, in the Ampere's law topic, but the sheet gives you the answer outright. So the solenoid is the one Ampere case you can quote rather than derive.

The cross product magnitude rule is not on this page. $\lvert \vec{A} \times \vec{B} \rvert = AB\sin\theta$ is printed on the fourth sheet, in the vectors block, alongside the dot product and the component form of vector addition. If you are doing Biot-Savart by hand and want the sine, that is where it lives; the vector operations reference collects that block.

Neither law appears on the AP Physics 2 sheet. That course prints $B = \frac{\mu_0 I}{2\pi r}$ as a bare result, which is the answer Ampere's law gives for a long wire, handed over without the law.

When it costs a mark

Trying Ampere's law on a loop or an arc. Writing $B(2\pi R) = \mu_0 I$ for the centre of a circular loop gives $\frac{\mu_0 I}{2\pi R}$, which is off from the correct $\frac{\mu_0 I}{2R}$ by a factor of $\pi$. It looks like a plausible answer, and the step that fails is the unstated claim that $B$ is constant and tangent along that circle.

Losing the direction because Biot-Savart's cross product was treated as a product. $d\vec{\ell} \times \hat{r}$ is a vector perpendicular to both. If you compute magnitudes only and never say which way the contributions point, you cannot know whether they add or cancel, and on the axis of a loop they partly cancel.

Confusing $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I(d\vec{\ell} \times \hat{r})}{r^2}$ with $\vec{F}_B = \int I(d\vec{\ell} \times \vec{B})$. One is the field a current makes; the other is the force a field puts on a current. Both are printed within four lines of each other on the sheet and both contain $I\,d\vec{\ell} \times$ something. Read which vector is inside the cross product.

Forgetting that $\hat{r}$ points from the source element to the field point. Reversing it flips the sign of every contribution, which flips the direction of the answer while leaving its magnitude right.

Using $n$ as the number of turns in the solenoid formula. In $B_{\mathrm{sol}} = \mu_0 n I$, the sheet's symbol list defines $n$ as the number of loops per unit length, while $N$ is the total number of loops. A solenoid with 500 turns over $0.25 \ \mathrm{m}$ has $n = 2000 \ \mathrm{m^{-1}}$, and using 500 understates the field by a factor of four.

Where these sit in the course

The Biot-Savart law is Topic 12.3 and Ampere's law is Topic 12.4, consecutive and closing topics of Unit 12, Magnetic Fields and Electromagnetism, which the CED weights at 10 to 20 percent of the multiple-choice section over roughly 10 to 20 class periods.

The ordering matches Unit 8's. There, the integral form of Coulomb's law comes in Topic 8.4 and Gauss's law arrives in 8.6 as the shortcut. Here, Biot-Savart comes in 12.3 and Ampere's law arrives in 12.4 as the shortcut. In both units the labour-intensive law is taught first, because the shortcut only makes sense once you know what it is short for.

Essential knowledge 12.3.A.2 is worth reading before either calculation: the magnetic field vectors around a small segment of current-carrying wire are tangent to concentric circles centred on that wire, and the field has no component toward, away from, or parallel to the segment. That single sentence is the geometric fact both laws are built on, and it is why a magnetic field can never be handled by a closed-surface argument the way an electric field can. The structural comparison is Gauss's law vs Ampere's law.

What comes next in the course uses the output of both. Unit 13 takes these fields, pushes them through a loop as magnetic flux, and asks what happens when the flux changes.