Question

Express the uniform vector field \(\mathbf{F}=5 \mathbf{a}_{x}\) in \((a)\) cylindrical components; (b) spherical components.

Short answer

Question: Express the vector field \(\mathbf{F}=5\mathbf{a}_x\) in (a) cylindrical components and (b) spherical components. Answer: (a) In cylindrical components, the vector field \(\mathbf{F}\) can be expressed as: \(\mathbf{F'} = 5(\cos{\theta} \, \mathbf{a}_{\rho} - \sin{\theta} \, \mathbf{a}_{\theta})\) (b) In spherical components, the vector field \(\mathbf{F}\) can be expressed as: \(\mathbf{F''} = 5(\sin{\phi} \cos{\theta} \, \mathbf{a}_{r} + \cos{\phi} \cos{\theta}\, \mathbf{a}_{\phi} - \sin{\theta}\, \mathbf{a}_{\theta})\)

Step-by-step solution

Separation of Cartesian Components

The given vector field \(\mathbf{F}\) is already expressed in Cartesian components: $$ \mathbf{F}=5 \mathbf{a}_{x}. $$

Express in Cylindrical Components

To convert into cylindrical components, we'll use the following transformation equations for the \(x\) component: $$\mathbf{a}_x = \cos{\theta} \, \mathbf{a}_\rho - \sin{\theta} \, \mathbf{a}_\theta$$ Thus, the vector field in cylindrical components can be expressed as: $$ \mathbf{F'} = 5 \mathbf{a}_x = 5(\cos{\theta} \, \mathbf{a}_\rho - \sin{\theta} \, \mathbf{a}_\theta) $$

Express in Spherical Components

To convert into spherical components, we'll use the following transformation equations for the \(x\) component: $$\mathbf{a}_x = \sin{\phi} \cos{\theta} \, \mathbf{a}_r + \cos{\phi} \cos{\theta}\, \mathbf{a}_\phi - \sin{\theta}\, \mathbf{a}_\theta$$ Then, the vector field in spherical components can be expressed as: $$ \mathbf{F''} = 5 \mathbf{a}_x = 5(\sin{\phi} \cos{\theta} \, \mathbf{a}_r + \cos{\phi} \cos{\theta}\, \mathbf{a}_\phi - \sin{\theta}\, \mathbf{a}_\theta) $$ So, the vector field \(\mathbf{F}\) is expressed in (a) cylindrical components as: $$ \mathbf{F'} = 5(\cos{\theta} \, \mathbf{a}_{\rho} - \sin{\theta} \, \mathbf{a}_{\theta}) $$ and (b) spherical components as: $$ \mathbf{F''} = 5(\sin{\phi} \cos{\theta} \, \mathbf{a}_{r} + \cos{\phi} \cos{\theta}\, \mathbf{a}_{\phi} - \sin{\theta}\, \mathbf{a}_{\theta}) $$

Key concepts

Cylindrical Components

The concept of cylindrical components allows us to transform a vector field expressed in Cartesian coordinates, like \( \mathbf{F} = 5 \mathbf{a}_x \), into a format better suited for scenarios where cylindrical symmetry is present, such as in pipes or along a cylinder.
Cylindrical coordinates consist of three components: radial \( (\rho) \), angular \( (\theta) \), and the height \( (z) \). The transformation requires mapping each Cartesian component to its cylindrical counterpart.
The conversion utilizes the formulas:

• \( \mathbf{a}_x = \cos{\theta} \, \mathbf{a}_\rho - \sin{\theta} \, \mathbf{a}_\theta \)
• \( \mathbf{a}_y = \sin{\theta} \, \mathbf{a}_\rho + \cos{\theta} \, \mathbf{a}_\theta \)
Using these relationships, the original vector field can be expressed in cylindrical components, providing insights and solutions that align with the geometry and physics of cylindrical shapes.

Spherical Components

When dealing with phenomena involving radial symmetry, such as electric fields around a charged sphere, spherical components come in handy. These unfold neatly in scenarios where vectors point inwards or outwards from a sphere's center.
Spherical coordinates describe a point in space with three components: radial \( (r) \), polar \( (\phi) \), and azimuthal \( (\theta) \). Transforming a vector from Cartesian to spherical coordinates involves understanding these axes:

• Radial \( (r) \): distance from the origin.
• Polar \( (\phi) \): angle from the positive z-axis.
• Azimuthal \( (\theta) \): angle from the positive x-axis in the xy-plane.

The conversion equation for the x-component, as seen, is:

• \( \mathbf{a}_x = \sin{\phi} \cos{\theta} \, \mathbf{a}_r + \cos{\phi} \cos{\theta}\, \mathbf{a}_\phi - \sin{\theta}\, \mathbf{a}_\theta \).

This transformation reshapes our vector field to better interact with spherical geometries.

Coordinate Systems

Understanding the different coordinate systems is crucial in expressing vectors in terms that match the geometry of a problem. Each system highlights symmetry or certain features, offering advantages in analysis and computation.

• **Cartesian Coordinates**: Best for flat, rectangular spaces. Use three perpendicular axes (x, y, z).
• **Cylindrical Coordinates**: Ideal for problems revolving around a central axis. They use radial \( (\rho) \), angular \( (\theta) \), and height \( (z) \) as dimensions.
• **Spherical Coordinates**: Perfect for radially symmetric situations, such as in celestial or atomic models, using radial \( (r) \), polar \( (\phi) \), and azimuthal \( (\theta) \) dimensions.

Switching between these systems, as seen in our vector field transformation exercise, allows for aligning mathematical problems with real-world geometries, facilitating easier and more intuitive solutions.

Cartesian Components

In the Cartesian coordinate system, vectors are expressed in terms of their x, y, and z components. This system is intuitive, using straight line distances along perpendicular axes.
For the given vector \( \mathbf{F}=5 \mathbf{a}_{x} \), it's clear and straightforward: the vector is along the x-axis, scaled by 5. This simple representation is best suited for problems in and straight equations in flat, rectangular domains.
Because Cartesian orientation does not easily align with circular or spherical shapes, transformations to other systems—like cylindrical or spherical—are necessary for real-world applications. These transformations re-express the vector components to match the natural symmetry of the scenarios and significantly aid in solving varied physics problems efficiently.