The electric potential in a region of space is V=100(x^2-....

The electric potential in a region of space is $V=100\left(x^{2}-\right.$ $y^{2}$ ) $V,$ where $x$ and $y$ are in meters. a. Draw a contour map of the potential, showing and labeling the $-400 \mathrm{V},-100 \mathrm{V}, 0 \mathrm{V},+100 \mathrm{V},$ and $+400 \mathrm{V}$ equipotential surfaces. b. Find an expression for the electric field $\vec{E}$ at position $(x, y)$ c. Draw the electric field lines on your diagram of part a.

The electric potential in a region of space is $V=100\left(x^{2}-\right.$ $y^{2}$ ) $V,$ where $x$ and $y$ are in meters.
a. Draw a contour map of the potential, showing and labeling the $-400 \mathrm{V},-100 \mathrm{V}, 0 \mathrm{V},+100 \mathrm{V},$ and $+400 \mathrm{V}$ equipotential surfaces.
b. Find an expression for the electric field $\vec{E}$ at position $(x, y)$
c. Draw the electric field lines on your diagram of part a.

Key Concepts

Electric Potential

Electric potential is a scalar quantity that represents the potential energy per unit charge at any point in space due to a source distribution. In problems involving charged fields, the electric potential function defines how energy values change with position, and its distribution often helps in understanding the behavior of test charges within the field.

Equipotential Surfaces

Equipotential surfaces are geometric loci where the electric potential is constant. They are significant because no work is done when moving a charge along these surfaces, and they provide useful visual aids for understanding the spatial variation of the electric potential. These surfaces help in illustrating how the potential changes and in locating the direction of the electric field lines, which are always perpendicular to them.

Electric Field

The electric field is a vector field that represents the force per unit charge experienced by a test charge at any point in space. It is directly related to the spatial variation of the electric potential, with the field being the negative gradient of the potential. Understanding the electric field is essential for predicting the motion of charges and for relating energy considerations to force interactions.

Gradient Operator

The gradient operator is a mathematical tool used to determine the rate and direction of change in a scalar field, such as electric potential. In electromagnetism, applying the gradient to the electric potential yields the electric field (with a negative sign), indicating both the magnitude and the direction in which the potential decreases most rapidly. This concept is fundamental to converting potential functions into field descriptions and provides a systematic way of analyzing field behavior across space.

Transcript

0:00 Okay.

00:01 So for this question, we know the potential function is 100 times x -square minus y -square voltage.

00:07 So when the potential is equal to negative 400 volts, we have x -square minus y -square is equal to negative -4.

00:13 Therefore, we have y -square over 4 minus x -square over 4 is equal to 1.

00:18 And this is the graph for y -square over 4 minus x -square over 4 equal to 1.

00:23 Okay? and then we know when the potential function is equal to negative 100 -volt, we have x -square minus y -square, to negative 1.

00:32 So therefore, y squared minus x squared is equal to 1.

00:34 And this is the graph for y square minus x squared is equal to 1.

00:41 And when the potential becomes 0, x squared minus y squared just become 0, ok? so we have x squared equal to y squared, which means that positive negative x is equal to positive negative y.

00:53 And this gives us four combinations here.

00:55 But eventually, it's actually just two combinations, which is y is equal to x and y is equal negative x.

01:00 So this is the graph for y is equal to x and y is equal to x.

01:03 Is equal to negative x.

01:07 And then let's plug in a hundred volt here.

01:10 So when we plug in 100 volt, we have x squared minus y squared is equal to one.

01:14 And this is the graph for x square minus y square, equal to one.

01:19 And when the potential is equal to 400 volts, we'll have x squared minus y square is equal to four...