Apparent Contradiction To Dispersion Relation ∣ K ∣ = Ω C |\textbf {k}|=\frac{\omega}{c} ∣ K ∣ = C Ω ​ Or Drift Velocity At Speed Of Light In Free Space?

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Apparent Contradiction to Dispersion Relation ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c} or Drift Velocity at Speed of Light in Free Space?

The speed of light in free space is a fundamental constant in physics, denoted by the letter c. It is a universal speed limit that no object or information can exceed. However, when it comes to electromagnetic (EM) waves, the situation becomes more complex. The dispersion relation, which describes the relationship between the wavevector (k) and the angular frequency (ω) of an EM wave, is given by ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c}. This equation suggests that the speed of an EM wave is equal to the speed of light in free space. However, this apparent contradiction arises when we consider the drift velocity of an EM wave, which seems to be equal to the speed of light in free space. In this article, we will explore this apparent contradiction and examine the underlying physics.

To understand the apparent contradiction, we need to start with Maxwell's equations for EM waves. These equations are a set of four fundamental equations that describe the behavior of EM fields in space and time. The four equations are:

∇⋅E⃗=0(1)\nabla \cdot \vec{E}=0 \tag{1}

∇⋅B⃗=0(2)\nabla \cdot \vec{B}=0 \tag{2}

∇×E⃗=−∂B⃗∂t(3)\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t} \tag{3}

∇×B⃗=μ0ϵ0∂E⃗∂t(4)\nabla \times \vec{B}=\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \tag{4}

where E⃗\vec{E} is the electric field, B⃗\vec{B} is the magnetic field, μ0\mu_0 is the magnetic constant, and ϵ0\epsilon_0 is the electric constant.

Using Maxwell's equations, we can derive the dispersion relation for EM waves. We start by taking the curl of equation (3):

∇×(∇×E⃗)=−∂∂t(∇×B⃗)\nabla \times (\nabla \times \vec{E})=-\frac{\partial}{\partial t} (\nabla \times \vec{B})

Using the vector identity ∇×(∇×A⃗)=∇(∇⋅A⃗)−∇2A⃗\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A}, we get:

∇(∇⋅E⃗)−∇2E⃗=−∂∂t(∇×B⃗)\nabla (\nabla \cdot \vec{E}) - \nabla^2 \vec{E}=-\frac{\partial}{\partial t} (\nabla \times \vec{B})

Since ∇⋅E⃗=0\nabla \cdot \vec{E}=0 (equation 1), we are left with:

−∇2E⃗=−∂∂t(∇×B⃗)-\nabla^2 \vec{E}=-\frac{\partial}{\partial t} (\nabla \times \vec{B})

Now, we take the curl of equation (4):

∇×(∇×B⃗)=μ0ϵ0∂∂t(∇×E⃗)\nabla \times (\nabla \times \vec{B})=\mu_0 \epsilon_0 \frac{\partial}{\partial t} (\nabla \times \vec{E})

Using the vector identity ∇×(∇×A⃗)=∇(∇⋅A⃗)−∇2A⃗\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A}, we get:

∇(∇⋅B⃗)−∇2B⃗=μ0ϵ0∂∂t(∇×E⃗)\nabla (\nabla \cdot \vec{B}) - \nabla^2 \vec{B}=\mu_0 \epsilon_0 \frac{\partial}{\partial t} (\nabla \times \vec{E})

Since ∇⋅B⃗=0\nabla \cdot \vec{B}=0 (equation 2), we are left with:

−∇2B⃗=μ0ϵ0∂∂t(∇×E⃗)-\nabla^2 \vec{B}=\mu_0 \epsilon_0 \frac{\partial}{\partial t} (\nabla \times \vec{E})

Now, we can substitute the expression for ∇×B⃗\nabla \times \vec{B} from equation (4) into the above equation:

−∇2B⃗=μ0ϵ0∂∂t(μ0ϵ0∂E⃗∂t)-\nabla^2 \vec{B}=\mu_0 \epsilon_0 \frac{\partial}{\partial t} \left(\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}\right)

Simplifying, we get:

−∇2B⃗=μ02ϵ02∂2E⃗∂t2-\nabla^2 \vec{B}=\mu_0^2 \epsilon_0^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Now, we can substitute the expression for ∇2E⃗\nabla^2 \vec{E} from the first equation into the above equation:

−∇2B⃗=−μ02ϵ02∂2E⃗∂t2-\nabla^2 \vec{B}=-\mu_0^2 \epsilon_0^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Simplifying, we get:

∇2B⃗=μ02ϵ02∂2E⃗∂t2\nabla^2 \vec{B}=\mu_0^2 \epsilon_0^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Now, we can substitute the expression for ∇2B⃗\nabla^2 \vec{B} from the above equation into the equation for ∇2E⃗\nabla^2 \vec{E}:

−∇2E⃗=−μ02ϵ02∂2E⃗∂t2-\nabla^2 \vec{E}=-\mu_0^2 \epsilon_0^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Simplifying, we get:

∇2E⃗=μ02ϵ02∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0^2 \epsilon_0^2 \frac{\partial^2 \vec{E}}{\partial t^2}

This is the wave equation for the electric field, which describes the propagation of EM waves.

To derive the dispersion relation, we need to consider the wave equation for the electric field in a medium with a constant refractive index n. The wave equation is given by:

∇2E⃗=μ0ϵ0n2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0 \epsilon_0 n^2 \frac{\partial^2 \vec{E}}{\partial t^2}

We can rewrite this equation as:

∇2E⃗=μ0ϵ0(nc)2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0 \epsilon_0 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

where c is the speed of light in free space.

Now, we can substitute the expression for ∇2E⃗\nabla^2 \vec{E} from the above equation into the equation for ∇2B⃗\nabla^2 \vec{B}:

∇2B⃗=μ02ϵ02(nc)2∂2E⃗∂t2\nabla^2 \vec{B}=\mu_0^2 \epsilon_0^2 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Simplifying, we get:

∇2B⃗=μ02ϵ02(nc)2∂2E⃗∂t2\nabla^2 \vec{B}=\mu_0^2 \epsilon_0^2 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Now, we can substitute the expression for ∇2B⃗\nabla^2 \vec{B} from the above equation into the equation for ∇2E⃗\nabla^2 \vec{E}:

∇2E⃗=μ02ϵ02(nc)2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0^2 \epsilon_0^2 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

Simplifying, we get:

∇2E⃗=μ02ϵ02(nc)2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0^2 \epsilon_0^2 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

This is the wave equation for the electric field in a medium with a constant refractive index n.

To derive the dispersion relation, we need to consider the wave equation for the electric field in a medium with a constant refractive index n. The wave equation is given by:

∇2E⃗=μ0ϵ0n2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0 \epsilon_0 n^2 \frac{\partial^2 \vec{E}}{\partial t^2}

We can rewrite this equation as:

∇2E⃗=μ0ϵ0(nc)2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0 \epsilon_0 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

where c is the speed of light in free space.

Now, we can substitute the expression for ∇2E⃗\nabla^2 \vec{E} from the above equation into the equation for ∇2B⃗\nabla^2 \vec{B}:

∇2B⃗=μ02ϵ02(nc)2∂2E⃗∂t2\nabla^2 \vec{B}=\mu_0^2 \epsilon_0^2 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

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Apparent Contradiction to Dispersion Relation ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c} or Drift Velocity at Speed of Light in Free Space?

Q: What is the apparent contradiction between the dispersion relation ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c} and the drift velocity of an EM wave?

A: The apparent contradiction arises when we consider the drift velocity of an EM wave, which seems to be equal to the speed of light in free space. However, the dispersion relation ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c} suggests that the speed of an EM wave is equal to the speed of light in free space.

Q: How do we derive the dispersion relation ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c}?

A: We can derive the dispersion relation by considering the wave equation for the electric field in a medium with a constant refractive index n. The wave equation is given by:

∇2E⃗=μ0ϵ0n2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0 \epsilon_0 n^2 \frac{\partial^2 \vec{E}}{\partial t^2}

We can rewrite this equation as:

∇2E⃗=μ0ϵ0(nc)2∂2E⃗∂t2\nabla^2 \vec{E}=\mu_0 \epsilon_0 \left(\frac{n}{c}\right)^2 \frac{\partial^2 \vec{E}}{\partial t^2}

where c is the speed of light in free space.

Q: What is the significance of the refractive index n in the dispersion relation?

A: The refractive index n is a measure of the speed of light in a medium. In a vacuum, the refractive index is equal to 1, and the speed of light is equal to c. In a medium with a refractive index n, the speed of light is reduced by a factor of n.

Q: How do we relate the drift velocity of an EM wave to the dispersion relation?

A: The drift velocity of an EM wave is given by:

vd=ωkv_d=\frac{\omega}{k}

where ω is the angular frequency and k is the wavevector. We can relate the drift velocity to the dispersion relation by noting that:

ωk=ω∣k∣\frac{\omega}{k}=\frac{\omega}{|\textbf{k}|}

Q: What is the apparent contradiction between the drift velocity and the dispersion relation?

A: The apparent contradiction arises when we consider the fact that the drift velocity of an EM wave seems to be equal to the speed of light in free space, while the dispersion relation suggests that the speed of an EM wave is equal to the speed of light in free space.

Q: How do we resolve the apparent contradiction?

A: We can resolve the apparent contradiction by noting that the drift velocity of an EM wave is not equal to the speed of light in free space. Instead, the drift velocity is related to the group velocity of the EM wave, which is given by:

vg=dωdkv_g=\frac{d\omega}{dk}

Q: What is the significance of the group velocity in the context of the apparent contradiction?

A: The group velocity is a measure of the speed at which the energy of an EM wave propagates through a medium. In the context of the apparent contradiction, the group velocity is related to the drift velocity of the EM.

Q: How do we relate the group velocity to the dispersion relation?

A: We can relate the group velocity to the dispersion relation by noting that:

vg=dωdk=dωd∣k∣v_g=\frac{d\omega}{dk}=\frac{d\omega}{d|\textbf{k}|}

Q: What is the significance of the derivative of the angular frequency with respect to the wavevector in the context of the apparent contradiction?

A: The derivative of the angular frequency with respect to the wavevector is a measure of the rate at which the energy of an EM wave propagates through a medium. In the context of the apparent contradiction, this derivative is related to the group velocity of the EM wave.

In conclusion, the apparent contradiction between the dispersion relation ∣k∣=ωc|\textbf{k}|=\frac{\omega}{c} and the drift velocity of an EM wave arises from a misunderstanding of the relationship between the drift velocity and the group velocity of the EM wave. By considering the group velocity and its relation to the dispersion relation, we can resolve the apparent contradiction and gain a deeper understanding of the behavior of EM waves in different media.