Zero Electric Field Point Calculation

Hey everyone! Today, we're diving into a fascinating problem in electromagnetism: figuring out where the electric field is zero between two charges. Let's say we have a -2 µC charge and an 8 µC charge sitting 3 meters apart. Our mission, should we choose to accept it, is to pinpoint the exact location where the electric field cancels out. Buckle up, because we're about to embark on an electrifying journey!

Understanding the Electric Field

Before we jump into calculations, let's quickly recap what the electric field actually is. The electric field is a vector field that surrounds an electric charge and exerts a force on other charges within its vicinity. Imagine it as an invisible force field radiating outwards (or inwards for negative charges). The electric field strength is determined by the magnitude of the charge creating the field and the distance from that charge. A larger charge creates a stronger field, and the field strength decreases as you move further away.

Now, consider our two charges: the -2 µC charge (negative) and the 8 µC charge (positive). Electric field lines point away from positive charges and towards negative charges. This means the electric field lines from the 8 µC charge will point away from it, while the field lines from the -2 µC charge will point towards it. The crucial point here is that somewhere between these charges, the electric fields created by them will oppose each other. This is where we have the potential for the fields to cancel out, resulting in a net electric field of zero.

Setting Up the Problem

To solve this, we'll use the principle of superposition, which states that the total electric field at a point is the vector sum of the electric fields due to all individual charges. In simpler terms, we'll calculate the electric field due to each charge separately and then add them up (considering their directions). We are trying to find the distance from the -2µC charge to the point where the electric field is zero.

Let's define our variables:

• q1 = -2 µC (charge 1)
• q2 = 8 µC (charge 2)
• d = 3 m (distance between the charges)
• x = distance from q1 (-2 µC) to the point where the electric field is zero. This is what we want to find.

Therefore, the distance from q2 (8 µC) to the zero electric field point will be d - x = 3 - x.

Calculating the Electric Fields

The electric field (E) created by a point charge (q) at a distance (r) is given by Coulomb's Law:

E = k * |q| / r²

where:

• k is Coulomb's constant (approximately 8.99 × 10⁹ N⋅m²/C²)
• |q| is the absolute value of the charge (we use the absolute value because we're only interested in the magnitude of the field, not its direction)
• r is the distance from the charge to the point of interest.

Now, let's apply this to our problem. Let E₁ be the electric field due to q1 (-2 µC) and E₂ be the electric field due to q2 (8 µC) at the point where the net electric field is zero.

The magnitude of E₁ at the point x from q1 is:

E₁ = k * |q1| / x² = k * |-2 × 10⁻⁶ C| / x² = k * 2 × 10⁻⁶ C / x²

The magnitude of E₂ at the point (3 - x) from q2 is:

E₂ = k * |q2| / (3 - x)² = k * |8 × 10⁻⁶ C| / (3 - x)² = k * 8 × 10⁻⁶ C / (3 - x)²

Finding the Zero Field Point

Remember, we're looking for the point where the electric field is zero. This means the magnitudes of E₁ and E₂ must be equal, but their directions must be opposite. Since q1 is negative and q2 is positive, their electric fields will indeed point in opposite directions between the charges. Therefore, we can set the magnitudes equal to each other:

E₁ = E₂

k * 2 × 10⁻⁶ C / x² = k * 8 × 10⁻⁶ C / (3 - x)²

Notice that Coulomb's constant (k) and the 10⁻⁶ C term appear on both sides of the equation. We can simplify by canceling them out:

2 / x² = 8 / (3 - x)²

Now we have a much simpler equation to solve. Let's cross-multiply:

2 * (3 - x)² = 8 * x²

Expanding the left side:

2 * (9 - 6x + x²) = 8x²

18 - 12x + 2x² = 8x²

Solving the Quadratic Equation

Now, let's rearrange the equation to get a standard quadratic form (ax² + bx + c = 0):

0 = 6x² + 12x - 18

We can simplify this further by dividing the entire equation by 6:

0 = x² + 2x - 3

Now we have a quadratic equation that we can solve by factoring:

(x + 3)(x - 1) = 0

This gives us two possible solutions for x:

• x = -3
• x = 1

Interpreting the Solutions

We have two solutions, but we need to figure out which one makes sense in our physical context. Remember, x represents the distance from the -2 µC charge to the point where the electric field is zero. The distance between the two charges is 3 meters, so the point where the electric field is zero must lie somewhere between the charges.

The solution x = -3 meters doesn't make sense because it implies the zero field point is 3 meters to the left of the -2 µC charge, which is outside the region between the charges. However, the solution x = 1 meter does make sense. It means the point where the electric field is zero is 1 meter away from the -2 µC charge and, consequently, 2 meters away from the 8 µC charge.

The Final Answer

Therefore, the electric field is zero at a point 1 meter from the -2 µC charge and 2 meters from the 8 µC charge.

Key Takeaways

• The electric field is a vector field that describes the force exerted on electric charges.
• The electric field created by a point charge is proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge.
• The principle of superposition allows us to calculate the total electric field due to multiple charges by adding their individual electric fields vectorially.
• When dealing with multiple charges, the electric field can be zero at certain points where the contributions from different charges cancel each other out.
• Solving for the zero electric field point often involves setting the magnitudes of the electric fields equal and solving a resulting equation, which may be a quadratic equation.

I hope this explanation was clear and helpful! Electromagnetism can seem a bit daunting at first, but by breaking down problems step-by-step, we can conquer even the trickiest challenges. Keep exploring, keep questioning, and keep learning!