Plz check this Q :-

Figure 1 is a graph of Ex, the x component of the electric field, versus position. Find the graph of V(x). [ Assume V=0 & V at x = 0 m ]

To find the graph of the electric potential \( V(x) \) from the electric field \( E_x \) versus position graph, we need to understand the relationship between electric field and electric potential. The electric field is related to the potential by the equation:

Understanding the Relationship

The electric field \( E \) is defined as the negative gradient of the electric potential \( V \). Mathematically, this is expressed as:

E_x = -\frac{dV}{dx}

This means that the electric field at a point is equal to the negative rate of change of the potential with respect to position. If we want to find the potential \( V(x) \), we can rearrange this equation:

\frac{dV}{dx} = -E_x

Finding the Potential from the Electric Field

To find \( V(x) \), we can integrate the electric field with respect to \( x \). The integration will give us the potential function, plus a constant of integration \( C \). The equation becomes:

V(x) = -\int E_x \, dx + C

Applying the Boundary Condition

In your case, you mentioned that \( V = 0 \) at \( x = 0 \, m \). This boundary condition will help us determine the constant \( C \). After performing the integration, we can substitute \( x = 0 \) into our potential function to solve for \( C \).

Steps to Create the Graph of V(x)

• Step 1: Identify the shape of the graph for \( E_x \) versus \( x \). Is it linear, constant, or does it have some curvature?
• Step 2: Integrate the electric field graphically or analytically. If the graph is linear, the area under the curve will be a triangle or rectangle, which simplifies the integration process.
• Step 3: Calculate the potential at various points by applying the integration results and the boundary condition.
• Step 4: Plot the resulting values of \( V(x) \) against \( x \) to create the graph.

Example Calculation

Let’s say the graph of \( E_x \) is a constant value of \( 5 \, \text{N/C} \) from \( x = 0 \) to \( x = 2 \, \text{m} \). The integration would look like this:

V(x) = -\int 5 \, dx = -5x + C

Now, applying the boundary condition \( V(0) = 0 \):

0 = -5(0) + C

This gives us \( C = 0 \). Therefore, the potential function becomes:

V(x) = -5x

For \( x = 2 \, \text{m} \), the potential would be:

V(2) = -5(2) = -10 \, \text{V}

Visualizing the Result

Now, if you were to plot this, you would see a straight line with a negative slope, starting from the origin (0,0) and going downwards as \( x \) increases. The slope of the line corresponds to the magnitude of the electric field.

By following these steps and understanding the relationship between electric field and potential, you can effectively derive the potential graph from the electric field graph. If you have a specific graph for \( E_x \), we can go through the integration process together to find \( V(x) \) accurately!