Question

The electric potential in a volume of space is given by \(V(x, y, z)=x^{2}+x y^{2}+y z .\) Determine the electric field in this region at the coordinate (3,4,5).

Short answer

Question: Determine the electric field at the coordinate (3,4,5) for the given electric potential function \(V(x, y, z) = x^2 + xy^2 + yz\). Answer: The electric field at the coordinate (3,4,5) is \(\vec{E} = - (22\hat{i} + 29\hat{j} + 4\hat{k})\).

Step-by-step solution

Determine the Partial Derivatives of V(x, y, z)

To find the gradient, we need to compute the partial derivatives of the electric potential function with respect to x, y, and z. These will give us the components of the electric field. For the x-component: \(\frac{\partial V}{\partial x} = \frac{\partial(x^2 + xy^2 + yz)}{\partial x} = 2x + y^2\) For the y-component: \(\frac{\partial V}{\partial y} = \frac{\partial(x^2 + xy^2 + yz)}{\partial y} = 2xy + z\) For the z-component: \(\frac{\partial V}{\partial z} = \frac{\partial(x^2 + xy^2 + yz)}{\partial z} = y\)

Plug in the Coordinates

Now, we will plug in the coordinates (3,4,5) into the partial derivatives we obtained in Step 1 to find the electric field components at that point. For the x-component: \(\frac{\partial V}{\partial x}(3,4,5) = 2(3) + (4)^2 = 6 + 16 = 22\) For the y-component: \(\frac{\partial V}{\partial y}(3,4,5) = 2(3)(4) + (5) = 24+5 = 29\) For the z-component: \(\frac{\partial V}{\partial z}(3,4,5) = 4\)

Calculate the Electric Field Vector

Now that we have the x, y, and z components of the electric field, we can write the electric field vector \(\vec{E}\): \(\vec{E} = -\nabla V(x, y, z) = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)\) Plugging in the values we found in Step 2: \(\vec{E} = - (22\hat{i} + 29\hat{j} + 4\hat{k})\)

Final Answer

We have now found the electric field at the coordinate (3,4,5) for the given electric potential function: \(\vec{E} = - (22\hat{i} + 29\hat{j} + 4\hat{k})\)

Key concepts

Electric Potential

Electric potential is a fundamental concept in electromagnetism, closely related to the concept of electric field. Simply put, it represents the potential energy per unit charge at a point in a field, which a charge would have due to the presence of other charges or electric fields. It's a scalar quantity, which means it has magnitude but no direction, and is usually denoted by the letter 'V'. The unit for electric potential is the volt (V).

In the context of the current exercise, the electric potential is described by a function, \( V(x, y, z) = x^2 + xy^2 + yz \), where each point in space is assigned a value based on its coordinates. To understand electric fields better, we can think of electric potential as a landscape of hills and valleys. The steeper the slope or gradient of this landscape, the stronger the electric field at that point.

Partial Derivatives in Physics

Partial derivatives play a significant role in physics, particularly in fields like electromagnetism and thermodynamics. They help us understand how a function changes with respect to one variable while keeping other variables constant. This is crucial when dealing with multivariable functions, as it allows us to isolate the influence of each variable.

For instance, the electric potential given by \( V(x, y, z) \) is a function of three variables. We find how this potential changes in the direction of each coordinate axis by calculating the partial derivative with respect to that coordinate. As seen in the solution, each partial derivative reveals how sensitive the electric potential is to changes in x, y, or z separately. These derivatives lay the foundation to calculate the electric field vector, which represents the force a charge would experience at any given point.

Gradient of a Scalar Field

The gradient is a critical mathematical tool, particularly when dealing with scalar fields like electric potential. It's a vector that points in the direction of the greatest rate of increase of the scalar field and has a magnitude corresponding to that rate of increase.

In the context of electric potential, the gradient helps us determine the electric field, which is a vector field. The electric field is the negative gradient of the electric potential, and it provides direction and magnitude of the force that a charge would feel. We use the del operator (\( abla \)) to denote the gradient, and when we apply this operator to the electric potential function, we obtain the components of the electric field by taking the negative of these components.

Coordinate Systems in Physics

Physics often requires different coordinate systems to simplify calculations and better represent the symmetries of a problem. Common coordinate systems include Cartesian (rectangular), cylindrical, and spherical systems. Each has its unit vectors and basis for determining position, velocity, or other vector quantities in space.

The current exercise uses the Cartesian coordinate system, denoted by (x, y, z). It's the most straightforward system for problems with right angles and no inherent symmetry that suggests the use of polar coordinates. Understanding coordinate systems is essential for proper handling of vector quantities, like the electric field calculated here, and for translating physical problems into mathematical language that can be solved effectively.