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What is the difference between these laws? Which law is more useful? When to use Ampere's law and when to use Biot-Savart law?

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In the context of introductory Electromagnetic theory, you can use Ampere's law when the symmetry of the problem permits i.e. when the magnetic field around an 'Amperian loop' is constant. eg: to find the magnetic field from infinite straight current carrying wire at some radial distance.

Biot-Savart law is the more brute force approach, you evaluate this integral when there is not enough symmetry to use Ampere's law. eg: to evaluate the magnetic field at some point along the axis of a current loop.

HyperPhysics has some great examples: Amperes law, Biot-Savart law.

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  • $\begingroup$ "to evaluate the magnetic field at some point along the axis" or even - off axis... Biot-Savart, being an integral over the current, works for any current distribution. $\endgroup$ – Floris May 10 '14 at 17:24
  • $\begingroup$ Good point, it was just an example of something one might come across in intro physics. $\endgroup$ – peanut_butter May 11 '14 at 2:26
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What is the difference between these laws?

One interesting difference is that the Biot-Savart law is more general than the Ampère law.

The Ampère law $$ \oint_\gamma \mathbf B\cdot d\mathbf s = \mu_0 I $$ is valid only when the flux of electric field through the loop $\gamma$ is constant in time; otherwise its rate of change (the displacement current) has to be added to the normal current on the right-hand side. (Maxwell realized this).

However, the Biot-Savart law $$ \mathbf B (\mathbf x)= \int \frac{\mu_0}{4\pi} \frac{\mathbf j \times (\mathbf x - \mathbf r)}{|\mathbf x - \mathbf r|^3}\,d^3\mathbf r $$ although originally formulated for static situations as well, is more general, for it is valid even if the electric flux changes in time, provided the electric field is given by gradient of potential. This happens for example when capacitor connected to battery with negligible self-inductance gets charged. A magnetic field around the capacitor does not obey Ampère law, but it is given by the above Biot-Savart formula.

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  • $\begingroup$ does this means that bio savart law is equavalent to maxwell's 4th equation? $\endgroup$ – Prem kumar Aug 9 '15 at 8:46
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    $\begingroup$ No, it is considered less general than the $\nabla\times\mathbf B$ equation. $\endgroup$ – Ján Lalinský Aug 9 '15 at 10:19
  • $\begingroup$ @JánLalinský this is not correct. They are actually equivalent in the static case. $\endgroup$ – Gonenc Dec 6 '18 at 22:52
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    $\begingroup$ @AbhinavDhawan that is good question, such examples are not very common. Biot-Savart law can break down in situations where electric field is not given by a gradient. For example, when current in a very long and thin solenoid increases rapidly, magnetic flux inside increases and induced electric field appears also outside the solenoid. Biot-Savart law predicts zero magnetic field outside, but when the electric field outside changes in time, Faraday-Maxwell's equation for EM induction predicts that curl of $\mathbf B$ is non-zero outside. That is only possible if $\mathbf B$ does not vanish. $\endgroup$ – Ján Lalinský Apr 13 '20 at 16:42
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    $\begingroup$ @AbhinavDhawan Electric field outside the solenoid will change in time if the magnetic flux changes nonlinearly. That happens when electric current accelerates (second derivative of $I$ is non-zero). So in this case the breakdown of B-S law happens when the current is moving in a way that produces radiation. $\endgroup$ – Ján Lalinský Apr 13 '20 at 16:48
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If we draw analogy biot-savart law is the columb's law and amperes law is gausses' law (but it is not exactly the gausses' law for magnetism) then, it becomes clear when to use ampere's law and when biot-savart law.

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    $\begingroup$ This really jsut changes the question to "what's the difference between Coulomb's and Gauss's laws?" $\endgroup$ – Kyle Kanos Dec 6 '18 at 22:43
  • $\begingroup$ Columb's law describes the electric field of discrete charges while gausses law describes the field of continuous charge distribution. In other words, gausses law is the general and more practical form of Columb's law. Columb's law describes a fact of nature while gausses law says how to apply it. Columb's law is a fundamental law while gausses law is a mathematical manipulation of the fact that Columb's law describes. $\endgroup$ – Quanta Ali Ahmed Jan Nov 24 '19 at 22:32

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