(where the symbols have their usual meanings)
Understanding Electromagnetic Wave Propagation
The question asks for the speed of an electromagnetic wave as it travels through a linear homogeneous medium. We need to relate this speed to the speed of light in a vacuum and the properties of the medium.
Key Concepts
- Speed of Light in Vacuum (c): The speed of electromagnetic waves in a vacuum is a universal constant, denoted by c. It is defined using the permittivity of free space ($ \varepsilon_0 $) and the permeability of free space ($ \mu_0 $) as: $$c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$$
- Speed of Electromagnetic Wave in a Medium (v): When an electromagnetic wave travels through a material medium, its speed changes. This speed, denoted by v, depends on the permittivity ($ \varepsilon $) and permeability ($ \mu $) of that medium. The formula for the speed in the medium is: $$v = \frac{1}{\sqrt{\varepsilon \mu}}$$
- Linear Homogeneous Medium: This implies that the properties of the medium (permittivity $ \varepsilon $ and permeability $ \mu $) are constant throughout the medium and do not change with the strength of the electric or magnetic fields.
Deriving the Formula
We want to express the speed v in the medium in terms of c and the properties of the vacuum and the medium. We start with the formula for v:
$$v = \frac{1}{\sqrt{\varepsilon \mu}}$$To introduce c, we can multiply and divide by $ \sqrt{\varepsilon_0 \mu_0} $:
$$v = \frac{1}{\sqrt{\varepsilon \mu}} \times \frac{\sqrt{\varepsilon_0 \mu_0}}{\sqrt{\varepsilon_0 \mu_0}}$$Rearranging the terms, we get:
$$v = \left( \frac{1}{\sqrt{\varepsilon_0 \mu_0}} \right) \times \left( \frac{\sqrt{\varepsilon_0 \mu_0}}{\sqrt{\varepsilon \mu}} \right)$$Recognizing that $ \frac{1}{\sqrt{\varepsilon_0 \mu_0}} $ is the speed of light in vacuum, c, we can substitute it:
$$v = c \times \sqrt{\frac{\varepsilon_0 \mu_0}{\varepsilon \mu}}$$Comparing with Options
Let's compare our derived formula, $ v = c \sqrt{\frac{\varepsilon_0 \mu_0}{\varepsilon \mu}} $, with the given options:
- Option 1: $ c \sqrt{\frac{\varepsilon_0 \mu_0}{\varepsilon \mu}} $ - This matches our derived formula.
- Option 2: $ c \sqrt{\frac{\varepsilon \mu}{\varepsilon_0 \mu_0}} $ - This is the inverse of the correct factor.
- Option 3: $ c \sqrt{\frac{\varepsilon_0 \mu}{\varepsilon \mu_0}} $ - This incorrectly mixes vacuum and medium constants.
- Option 4: $ c \sqrt{\frac{\varepsilon \mu_0}{\varepsilon_0 \mu}} $ - This also incorrectly mixes constants.
Conclusion
The speed of an electromagnetic wave in a linear homogeneous medium is correctly given by the formula derived from the fundamental constants and the medium's properties.
$$v = c \sqrt{\frac{\varepsilon_0 \mu_0}{\varepsilon \mu}}$$This expression shows how the speed is reduced from its vacuum value c due to the presence of the medium's permittivity $ \varepsilon $ and permeability $ \mu $.
Important Questions from Electromagnetic Waves
- The velocity of ground-penetrating radar signals depends on
