Math Problem Statement
using the image, find the x-component of the electric field at the origin, point O.
Solution
To find the x-component of the electric field at the origin (O) due to the two point charges, we need to apply Coulomb's law and consider the contributions from both charges.
Given Data:
- at
- at
Coulomb's Law gives the magnitude of the electric field due to a point charge: where:
- is the electric field,
- is the Coulomb constant,
- is the charge,
- is the distance from the charge to the point of interest.
Step 1: Electric Field Due to
The distance from to the origin is . Using Coulomb's law: The electric field due to points toward the negative x-direction because the field direction is away from the positive charge at .
Step 2: Electric Field Due to
The distance from to the origin is . Using Coulomb's law: The electric field due to points toward the positive x-direction because the field direction is toward the negative charge at .
Step 3: Combine the Electric Fields
Now we sum the x-components of the electric fields. Both fields are along the x-axis, but they point in opposite directions:
- points in the negative x-direction (so it contributes ),
- points in the positive x-direction (so it contributes ).
Thus, the total x-component of the electric field at the origin is:
Final Answer:
The x-component of the electric field at the origin is 3.85 N/C in the positive x-direction.
Would you like any additional details or have further questions?
Here are 5 related questions to deepen your understanding:
- How would the result change if both charges were negative?
- What would the y-component of the electric field be at the origin?
- How does the distance between the charges affect the electric field strength?
- What would happen if a third charge were placed at the origin?
- How would you calculate the force on a third charge placed at the origin?
Tip: Always keep track of the direction of electric field vectors, as their direction is crucial in determining the net field.
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Math Problem Analysis
Mathematical Concepts
Electric Fields
Coulomb's Law
Vector Addition
Formulas
E = k_e * |q| / r^2
E_x = E_2 - E_1
Theorems
Coulomb's Law
Suitable Grade Level
Undergraduate Physics
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