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The electric potetnial in a region along...

The electric potetnial in a region along x-axis varies with `x` according to the releation `V(x) = 4 xx 5x^(2)`. Then the incorrect statement is

A

potential difference between the points x = 1 and x = -2 is 15 V.

B

the force experienced by the above charge will be towards the +x-axis.

C

a uniform electric field exists in this region along the +x-axis.

D

force experienced by a 1 C charge at x = -1 will be 10 N.

Text Solution

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The correct Answer is:
To solve the problem, we need to analyze the given electric potential function \( V(x) = 4 \times 5x^2 \) and determine which statements regarding this potential are incorrect. ### Step 1: Calculate the Electric Potential at Specific Points 1. **Calculate \( V(1) \)**: \[ V(1) = 4 \times 5 \times (1^2) = 20 \, \text{volts} \] 2. **Calculate \( V(-2) \)**: \[ V(-2) = 4 \times 5 \times ((-2)^2) = 4 \times 5 \times 4 = 80 \, \text{volts} \] ### Step 2: Find the Potential Difference 3. **Calculate the potential difference \( \Delta V \)**: \[ \Delta V = V(1) - V(-2) = 20 - 80 = -60 \, \text{volts} \] ### Step 3: Determine the Electric Field 4. **Calculate the Electric Field \( E \)**: The electric field \( E \) is given by the negative gradient of the potential: \[ E = -\frac{dV}{dx} \] First, we find \( \frac{dV}{dx} \): \[ V(x) = 20x^2 \quad \Rightarrow \quad \frac{dV}{dx} = 40x \] Therefore, \[ E = -40x \] ### Step 4: Analyze the Force on a Charge 5. **Calculate the force on a charge \( Q \)**: If we consider a charge \( Q = 1 \, \text{C} \) at \( x = -1 \): \[ E(-1) = -40 \times (-1) = 40 \, \text{N/C} \] The force \( F \) on the charge is given by: \[ F = Q \cdot E = 1 \cdot 40 = 40 \, \text{N} \] ### Step 5: Evaluate Statements Now we evaluate the statements based on our calculations: - **Statement A**: "The potential difference is 60." (Incorrect, as we found it to be -60) - **Statement B**: "The force experienced by a positive charge will be towards the positive x-axis." (Incorrect, as it will be towards the negative x-axis) - **Statement C**: "A uniform electric field exists in this region along the positive x-axis." (Incorrect, as the electric field is not uniform) - **Statement D**: "The force experienced by a one coulomb charge at \( x = -1 \) will be 40 N." (Correct) ### Conclusion The incorrect statements are A, B, and C.
To solve the problem, we need to analyze the given electric potential function \( V(x) = 4 \times 5x^2 \) and determine which statements regarding this potential are incorrect. ### Step 1: Calculate the Electric Potential at Specific Points 1. **Calculate \( V(1) \)**: \[ V(1) = 4 \times 5 \times (1^2) = 20 \, \text{volts} \] ...
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