Maxwell was the first to note that Amp\`ere's Law does not satisfy conservation of charge (his corrected form is given in Maxwell's equation). This can be shown using the equation of conservation of electric charge:
![{\displaystyle \nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27ae2865e26471949f2b0542781a15c2b6d2ec04)
Now consider Faraday's Law in differential form:
![{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb118e22c941e34f5537dbbdcaa3d7ba23603e0)
Taking the curl of both sides:
![{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla \times (-{\frac {\partial \mathbf {B} }{\partial t}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6dcc91d14635e28007ecd8879be3a8a1348976)
The right-hand side may be simplified by noting that
![{\displaystyle \nabla \times ({\frac {\partial \mathbf {B} }{\partial t}})=-{\frac {\partial }{\partial t}}(\nabla \times \mathbf {B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0f97040688483f243cba13e5435804d7daad4f)
Recalling Amp\`ere's Law,
![{\displaystyle -{\frac {\partial }{\partial t}}(\nabla \times \mathbf {B} )=-\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7b8fbc04615a600f8dc75fcc95bbb75ac7f3ad)
Therefore
![{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=-\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c54ce7f09ebc87f989b58f2fa312af7192d2513)
The left hand side may be simplified by the following Vector Identity:
![{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=-\nabla ^{2}\mathbf {E} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b087355bd418a7ec118691728a64e3de681b098)
Hence
![{\displaystyle \nabla ^{2}\mathbf {E} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93f91efe9424d50b0d6049ab1277d275c05f394e)
Applying the same analysis to Amp\'ere's Law then substituting in Faraday's Law leads to the result
![{\displaystyle \nabla ^{2}\mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b1554e300abc5edb662c24be970735163575e4)
Making the substitution
we note that these equations take the form of a transverse wave travelling at constant speed
. Maxwell evaluated the constants
and
according to their known values at the time and concluded that
was approximately equal to 310,740,000
, a value within ~3\