Question

In an EM wave, electric field amplitude is \( 300\,\text{V/m} \). Magnetic field amplitude is:

• A. $1 \times 10^{-6}\text{ T}$
• B. $1 \times 10^{-5}\text{ T}$
• C. $1 \times 10^{-7}\text{ T}$
• D. $3 \times 10^{-6}\text{ T}$

Hint:

The magnetic field vector in an EM wave is always much smaller in numerical value than the electric field vector because of the factor of $c$ ($3 \times 10^8$) in the denominator.
Always express your final answer in standard scientific notation to easily match the options.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Electromagnetic (EM) waves are transverse waves consisting of oscillating electric (\( E \)) and magnetic (\( B \)) fields.
A fundamental discovery of Maxwell's equations is that in a vacuum or in air, these two fields are inextricably linked.
The electric and magnetic fields oscillate in phase with each other and are always perpendicular to each other and to the direction of propagation.
Furthermore, the ratio of their magnitudes at any point in space and time is a constant.
This constant is defined by the speed of light (\( c \)), which represents the velocity at which these disturbances travel through a vacuum.
Step 2: Key Formula or Approach:
The relationship between the amplitude of the electric field (\( E_0 \)) and the amplitude of the magnetic field (\( B_0 \)) is given by:
\[ c = \frac{E_0}{B_0} \]
Where \( c \) is the speed of light in a vacuum (\( \approx 3 \times 10^8 \) m/s).
To find the magnetic field amplitude, we rearrange the formula to:
\[ B_0 = \frac{E_0}{c} \]
Step 3: Detailed Explanation:
In the given problem, we have:
- Electric field amplitude (\( E_0 \)) = \( 300 \) V/m.
- Speed of light (\( c \)) = \( 3 \times 10^8 \) m/s.
We substitute these values into our rearranged equation:
\[ B_0 = \frac{300}{3 \times 10^8} \]
To simplify this calculation, let's express \( 300 \) in scientific notation:
\[ 300 = 3 \times 10^2 \]
Now, perform the division:
\[ B_0 = \frac{3 \times 10^2}{3 \times 10^8} \]
The factor of \( 3 \) cancels out:
\[ B_0 = 1 \times 10^{2 - 8} \]
\[ B_0 = 1 \times 10^{-6} \text{ T} \]
The unit for magnetic field strength is the Tesla (T).
It is important to note how much smaller the magnetic field's numerical value is compared to the electric field.
Even though \( 1 \times 10^{-6} \) T seems small, in the context of an EM wave, it carries exactly the same amount of energy density as the \( 300 \) V/m electric field.
The energy is shared equally between the two fields because energy density is proportional to the square of the field strength, and the constants \( \epsilon_0 \) and \( \mu_0 \) account for the difference in magnitudes.
Comparing our result to the options, we find that it matches option (A).
Step 4: Final Answer:
The magnetic field amplitude of the EM wave is \( 1 \times 10^{-6} \) T.