In a certain region of space, the electric field is along `+y`-direction and has a magnitude of 400 V/m . What is the potential difference from the coordinate origin to the following points?
`a. x=0,y=20cm, z=0`
b. `x=0, y=-30cm, z=0`
`c. x=0, y=0, z=15cm`

Explanation

To solve the problem, we need to find the potential difference from the origin to the specified points in an electric field directed along the positive y-axis with a magnitude of 400 V/m. The potential difference \( V_O - V_P \) can be calculated using the formula:

\[ V_O - V_P = -E \cdot D \]

where \( E \) is the electric field strength and \( D \) is the distance moved in the direction of the electric field.

### Step-by-Step Solution:

**Step 1: Calculate the potential difference for point a (x=0, y=20 cm, z=0)**

1. Convert the distance from centimeters to meters:
\[ y = 20 \text{ cm} = 0.2 \text{ m} \]

2. The distance \( D \) from the origin to point P is \( 0.2 \text{ m} \) in the positive y-direction.

3. Substitute the values into the formula:
\[ V_O - V_P = -E \cdot D = -400 \text{ V/m} \cdot 0.2 \text{ m} \]

4. Calculate:
\[ V_O - V_P = -80 \text{ V} \]

**Answer for part a:** The potential difference from the origin to the point (0, 20 cm, 0) is -80 V.

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**Step 2: Calculate the potential difference for point b (x=0, y=-30 cm, z=0)**

1. Convert the distance from centimeters to meters:
\[ y = -30 \text{ cm} = -0.3 \text{ m} \]

2. The distance \( D \) from the origin to point Q is \( 0.3 \text{ m} \) in the negative y-direction.

3. Substitute the values into the formula:
\[ V_O - V_Q = -E \cdot D = -400 \text{ V/m} \cdot (-0.3 \text{ m}) \]

4. Calculate:
\[ V_O - V_Q = 120 \text{ V} \]

**Answer for part b:** The potential difference from the origin to the point (0, -30 cm, 0) is 120 V.

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**Step 3: Calculate the potential difference for point c (x=0, y=0, z=15 cm)**

1. Convert the distance from centimeters to meters:
\[ z = 15 \text{ cm} = 0.15 \text{ m} \]

2. The point R is located in the z-direction, which is perpendicular to the electric field direction (y-direction).

3. Since the electric field is perpendicular to the displacement, the potential difference is:
\[ V_O - V_R = -E \cdot D = -400 \text{ V/m} \cdot 0.15 \text{ m} \cdot \cos(90^\circ) \]

4. Since \( \cos(90^\circ) = 0 \):
\[ V_O - V_R = 0 \]

**Answer for part c:** The potential difference from the origin to the point (0, 0, 15 cm) is 0 V.

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