The flux density at a point in space is given by B = 4xaₓ + 2kya${}_y$ + 8a${}_z$ Wb/m² . The value of constant k must be equal to

Magnetic Flux Density Field Analysis

The question asks us to find the value of a constant, k, within a given magnetic flux density vector field. The magnetic flux density, denoted by $\mathbf{B}$, describes the strength and direction of a magnetic field. A fundamental property of all magnetic fields in physics is that they are solenoidal. This means that the magnetic field lines do not start or end at any point; they always form closed loops.

Understanding the Solenoidal Property

Mathematically, the solenoidal property is expressed using the divergence theorem. For any magnetic flux density field $\mathbf{B}$, its divergence must be zero:

$$\nabla \cdot \mathbf{B}=0$$

This equation is one of Maxwell's equations and is crucial for solving problems involving magnetic fields. We will use this principle to find the value of k.

Step 1: Identify the Components of B

The given magnetic flux density vector is:

$$\mathbf{B}=4x{\mathbf{a}}_x+2ky{\mathbf{a}}_y+8{\mathbf{a}}_z{\text{Wb/m}}^2$$

From this vector equation, we can identify the components of the magnetic flux density field:

- The x-component is $B_x=4x$
- The y-component is $B_y=2ky$
- The z-component is $B_z=8$

Step 2: Calculate the Divergence of B

The divergence of a vector field $\mathbf{A}=A_x{\mathbf{a}}_x+A_y{\mathbf{a}}_y+A_z{\mathbf{a}}_z$ in Cartesian coordinates is given by:

$$\nabla \cdot \mathbf{A}=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$$

Applying this formula to our magnetic flux density field $\mathbf{B}$:

$$\nabla \cdot \mathbf{B}=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}$$

Step 3: Compute the Partial Derivatives

Now, we compute the partial derivative of each component with respect to its corresponding coordinate:

- Derivative of $B_x$ with respect to x: $$\frac{\partial B_x}{\partial x}=\frac{\partial (4x)}{\partial x}=4$$
- Derivative of $B_y$ with respect to y: $$\frac{\partial B_y}{\partial y}=\frac{\partial (2ky)}{\partial y}=2k$$
- Derivative of $B_z$ with respect to z: $$\frac{\partial B_z}{\partial z}=\frac{\partial (8)}{\partial z}=0$$ (The derivative of a constant is zero)

Step 4: Solve for the Constant k

Substitute the computed partial derivatives back into the divergence equation and set it equal to zero, based on the solenoidal property ($\nabla \cdot \mathbf{B}=0$):

$$4+2k+0=0$$

Now, we solve this equation for k:

$$4+2k=0$$

$$2k=-4$$

$$k=\frac{-4}{2}$$

$$k=-2$$

Conclusion

Therefore, the value of the constant k that satisfies the condition for a magnetic flux density field is -2.