Finding a Zero-Field Point | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

Finding a Zero-Field Point

Between two masses, the fields point in opposite directions. The net field is zero where their sizes are equal.

The zero-field point is not usually the midpoint. It lies closer to the smaller mass.

Find a zero-field point

Let the two masses be $M_1$ and $M_2$ , with separation d. Let P be x from $M_1$ , so it is $d-x$ from $M_2$ .
At P, the net field is zero only when the two opposite field sizes are equal.
$\dfrac{GM_1}{x^2}=\dfrac{GM_2}{(d-x)^2}$
The constant G cancels. The zero point is closer to the smaller mass. A smaller mass needs a shorter distance to give the same field strength.

Worked example

Find the point where the net field is zero

Two masses are 3.0 × 10⁸ m apart. Their masses are 4.0 × 10²⁴ kg and 1.0 × 10²⁴ kg. Find the zero-field point between them.

Let x be the distance from the larger mass.
$\dfrac{4.0}{x^2}=\dfrac{1.0}{(3.0\times10^8-x)^2}$
Both distances are positive, so take the positive square root.
$\dfrac{2}{x}=\dfrac{1}{3.0\times10^8-x}$
Solve for x. The result is measured from the larger mass.
$2(3.0\times10^8-x)=x$
$x=2.0\times10^8\ \text{m}$
This point is 1.0 × 10⁸ m from the smaller mass.

Answer

2.0\times10^8\ \text{m}

This distance is measured from the larger mass.

Common mistake

Do not use x for both distances. If P is x from the first mass, its distance from the second mass is

d-x

Draw both field arrows. At a zero-field point they point in opposite directions and have equal sizes. The point is closer to the smaller mass.