Problem 7 - Arihant AIEEE Physics

Question

The temperature of a point in space is given by \(T=\left(x^{2}+y^{2}-z^{2}\right) .\) A mosquito located at \((1,1,2)\) desires to fly in such a direction that it will get heat as soon as possible. Then unit vector is : (b) \(-\frac{2}{3} \hat{1}-\frac{2}{3} \hat{j}+\frac{1}{3}\) (i) \(\frac{2}{3} \hat{i}-\frac{2}{3} \hat{j}-\frac{1}{3}\) (d) none of these

Step-by-Step Solution

Answer

The unit vector is option (i): \(\frac{2}{3} \hat{i} - \frac{2}{3} \hat{j} - \frac{1}{3}\).

Step 1: Calculate the Gradient

The gradient of a function \(T\) in three dimensions is given by \(abla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right)\). For \(T=x^2+y^2-z^2\), compute the partial derivatives: \(\frac{\partial T}{\partial x} = 2x\), \(\frac{\partial T}{\partial y} = 2y\), and \(\frac{\partial T}{\partial z} = -2z\). Thus, the gradient is \(abla T = (2x, 2y, -2z)\).

Step 2: Evaluate the Gradient at the Given Point

Evaluate the gradient at the point \((1,1,2)\). Substitute \(x=1\), \(y=1\), and \(z=2\) into \(abla T = (2x, 2y, -2z)\) to obtain \(abla T = (2 \times 1, 2 \times 1, -2 \times 2) = (2, 2, -4)\).

Step 3: Normalize the Gradient to Find the Direction

To find the unit vector in the direction of the gradient, divide the gradient by its magnitude. The magnitude of \((2, 2, -4)\) is \(\sqrt{2^2 + 2^2 + (-4)^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6}\). Thus, the unit vector is \(\left(\frac{2}{2\sqrt{6}}, \frac{2}{2\sqrt{6}}, \frac{-4}{2\sqrt{6}}\right)\)."

Step 4: Simplify the Unit Vector

Simplifying \(\left(\frac{2}{2\sqrt{6}}, \frac{2}{2\sqrt{6}}, \frac{-4}{2\sqrt{6}}\right)\) gives \(\left(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}}\right)\). Multiplying by \(\frac{\sqrt{6}}{\sqrt{6}}\), we get \(\left(\frac{\sqrt{6}}{6}, \frac{\sqrt{6}}{6}, \frac{-2\sqrt{6}}{6}\right)\), which matches option (i): \(\frac{2}{3} \hat{i} - \frac{2}{3} \hat{j} - \frac{1}{3}\).

Key Concepts

Vector Calculus

Vector calculus is a field of mathematics that deals with vector fields and differentiable functions in various dimensions. It is essential for understanding physical phenomena such as electromagnetic fields, fluid dynamics, and in this case, temperature distributions. In a nutshell, vector calculus helps us describe how a scalar or vector field changes in space.

Consider the temperature distribution represented by a function of coordinates, such as our temperature function: \( T = x^2 + y^2 - z^2 \). This scalar function can be analyzed using the gradient, divergence, and curl. These operations help describe how the function changes in space, a core idea in vector calculus. It provides insight into the behavior of physical quantities.

• The gradient tells us the direction and rate of fastest increase of a scalar field.
• Divergence measures how much a field spreads out from a particular point.
• Curl describes the rotation or swirling of a field around a point.

In solving the mosquito's problem, we focused on the gradient, which plays a pivotal role in determining the direction where the temperature increases most rapidly. This concept is not only vital in this specific problem but is also a key aspect of many physical and engineering applications.

Partial Derivative

A partial derivative is a derivative where we hold all but one of the variables constant to observe how a function changes with respect to a single variable. This concept is crucial in understanding functions of multiple variables, like those used in physics to describe multi-dimensional dynamics.

Consider our function \( T = x^2 + y^2 - z^2 \). When we take the partial derivative of \( T \) with respect to \( x \), denoted \( \frac{\partial T}{\partial x} \), we treat \( y \) and \( z \) as constants. Here, it simplifies to \( 2x \). This process is repeated for each variable:

• \( \frac{\partial T}{\partial x} = 2x \)
• \( \frac{\partial T}{\partial y} = 2y \)
• \( \frac{\partial T}{\partial z} = -2z \)

The partial derivatives tell us how the temperature changes in a particular direction. By combining them, we form the gradient, which gives us comprehensive information on how the temperature changes in space. This method is fundamental in vector calculus, allowing us to capture the full details of a multi-variable system.

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates direction. It is not concerned with size or length but aims to provide a pure direction. In physics and mathematics, unit vectors are essential for specifying directions within the coordinate system.

When solving the mosquito's dilemma, derive the unit vector in the direction of the temperature gradient to identify the fastest warming path. First, compute the gradient \( (2, 2, -4) \) at the point \( (1,1,2) \). The next step is normalizing the vector - dividing each component by the vector's magnitude - to get a unit vector:

• Calculate the magnitude: \( \sqrt{2^2 + 2^2 + (-4)^2} = \sqrt{24} \)
• Normalize: \( \left( \frac{2}{\sqrt{24}}, \frac{2}{\sqrt{24}}, \frac{-4}{\sqrt{24}} \right)\)
• Simplify to: \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}} \right) \)

Finally, apply algebra to organize it further and validate as the probable direction towards rising temperature, provided as option (i). Understanding unit vectors allows navigation within a space, focusing purely on direction while shedding magnitude.

Temperature Gradient

The temperature gradient is a vector that describes the direction and rate at which temperature changes most rapidly. It is the application of the gradient concept to a temperature field, like our function \( T = x^2 + y^2 - z^2 \).

In simple terms, the temperature gradient points in the direction where the temperature increases the fastest. For the mosquito wanting to warm up quickly, flying along the temperature gradient ensures it moves towards higher temperatures. To get the gradient, we compute the partial derivatives for each variable and combine, resulting in a vector field:

• Calculated gradient: \( (2x, 2y, -2z) \)
• Evaluate at \( (1, 1, 2) \): \( (2, 2, -4) \)

The real utility comes with converting it to a unit vector that reveals the precise shortest path to higher temperatures. Thus, the temperature gradient is indispensable in studying thermal processes and material science, aiding predictions of heat flow, diffusion, and related phenomena.