Intensity And The Poynting Vector

Intensity measures the average power crossing unit area and equals the time average of the Poynting vector \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}. The instantaneous magnitude is S = \frac{EB}{\mu_0} = \frac{E^2}{\mu_0 c}, using B = E/c. Averaging over a cycle replaces E^2 by E_0^2/2, giving I = \frac{E_0^2}{2\mu_0 c}. Since \frac{1}{\mu_0 c} = c\varepsilon_0 (because c^2 = 1/\mu_0\varepsilon_0), this simplifies neatly to I = \frac{1}{2}c\varepsilon_0 E_0^2. The option c\varepsilon_0 E_0^2 omits the one-half from time averaging and would be the peak, not average, intensity. The form E_0 B_0/\mu_0 leaves out the averaging factor of one-half entirely. The expression \varepsilon_0 E_0^2/c has the wrong placement of c and gives incorrect dimensions for intensity. This derivation is the standard route used in the NCERT and JEE syllabus to relate intensity to field amplitude. The same intensity can equivalently be written as I = \langle u \rangle c, the average energy density carried forward at the speed of light, which is a useful cross-check that the factor of one-half and the speed c both appear exactly once. Recognising these equivalent forms helps avoid the common slip of either dropping the averaging factor or misplacing c. Dimensionally, c\varepsilon_0 E_0^2 yields watts per square metre, confirming the chosen expression is correct.