91Ó°ÊÓ

Electric potential is a measure of the electric potential energy per unit charge at a specific point in space. In simpler words, it's like the height for electric charge - just as height determines potential energy for gravity, electric potential determines potential energy for electric forces.
In the given exercise, the electric potential is given by the equation \( V= (2.0 \, \mathrm{V/m^2}) x^2 - (3.0 \, \mathrm{V/m^2}) y^2 \).
This means that at any point \((x, y)\) on the plane, you can determine the electric potential by plugging in the \(x\) and \(y\) coordinates into the formula. The unit \(\mathrm{V/m^2}\) indicates the variation of potential with respect to distance squared.
The properties of this function show how electric potential varies within space and eventually, how it can influence electric field calculations.

The gradient of a potential is a crucial concept because it helps us understand the direction and magnitude of the electric field derived from the potential.
The gradient of a scalar field, like electric potential \(V\), gives us a vector field, which in this case is the electric field \(\vec{E}\). Mathematically, this is expressed as \(\vec{E} = -abla V\), where \(abla\) (nabla) symbol denotes the gradient operator.
The negative sign indicates that the electric field points in the direction where the potential decreases the fastest. This corresponds to the natural behavior of charges moving from higher to lower potential regions, akin to objects falling downhill toward lower gravitational potential.

Partial derivatives are mathematical tools used when we deal with functions of multiple variables. They let us see how the function changes as one variable changes, while keeping other variables constant.
In the context of this problem, our task is to compute partial derivatives of the potential \(V\) with respect to \(x\) and \(y\). These partial derivatives help find the components of the electric field.
For example:

• For the x-component, compute \(E_x = -\frac{\partial V}{\partial x}\).
• For the y-component, compute \(E_y = -\frac{\partial V}{\partial y}\).

By calculating these derivatives at specific points, you get the values of the electric field components at that location, revealing the influence each coordinate has on the electric field.

Unit-vector notation is a compact and clear way to express vector quantities, like electric fields, by breaking them into direction components.
In our exercise, the electric field \(\vec{E}\) is expressed as \(\vec{E} = (-12.0 \hat{i} + 12.0 \hat{j}) \, \mathrm{V/m}\). Here:

• \(\hat{i}\) is the unit vector in the x-direction (horizontal).
• \(\hat{j}\) is the unit vector in the y-direction (vertical).

The numbers next to these unit vectors describe the magnitude of the electric field along each direction, providing a clear picture of how the field is oriented in space. This notation becomes immensely useful for vector addition and helps in visualizing directionality in multidimensional problems.