Physics 9702/42 — May/June 2011

Background Concept

A field of force is a way of describing “action at a distance”. Instead of thinking of two objects needing to touch to exert a force, we say one object creates a field in the space around it.

A second (test) object placed in that region experiences a force because it is in the field. Common examples are:

• gravitational field: force acts on mass
• electric field: force acts on charge

Understanding the Question

The question asks for a general definition of a field of force (not specifically gravitational or electric). It wants the idea of a region of space where an object would feel a force even without contact.

Approach

State the definition in one sentence: “region of space” + “force on a suitable object placed there” + “no contact required”.

Step-by-Step Reasoning

• Mention region of space (this distinguishes a field from just a force).
• Mention that a body placed in this region experiences a force.
• Imply the force is present without physical contact (key field idea).

Key Takeaways

• Fields model forces that act through space.
• A field is defined by what it does to a suitable test object placed in it.

Common Mistakes

• Defining it as “a force” rather than “a region of space where a force acts”.
• Forgetting the “placed in the region” idea (the field exists in space, not only when the second object is present).

Things to Be Careful About

• Keep the definition general; do not restrict it only to gravity or only to electricity unless asked.
• Do not confuse “field” with “field strength” (force per unit mass/charge).

Background Concept

Gravitational and electric fields are both long-range fields that can be described by inverse-square laws for point sources:

• Gravity between point masses:

$$F_g = \frac{G m_1 m_2}{r^2}$$

• Electric force between point charges:

$$F_e = \frac{1}{4\pi\varepsilon_0}\frac{Q_1 Q_2}{r^2}$$

The field idea is then:

• gravitational field strength $g$ is force per unit mass
• electric field strength $E$ is force per unit positive charge

Understanding the Question

You must state:

• one similarity (something true for both types of field)
• one difference (something that distinguishes them)

Many answers are possible; you only need one of each, but they must be clear and unambiguous.

Approach

Pick a robust similarity and difference that are always true:

• similarity: both act through space and for point sources follow an inverse-square dependence with distance
• difference: gravity is always attractive; electric interaction depends on the sign of charge and can repel as well as attract

Step-by-Step Reasoning

Similarity

• Both are “action at a distance” forces: a mass/charge produces a field around it.
• For isolated point objects, the force magnitude decreases as $1/r^2$.

Difference

• Mass is always positive, so gravitational interaction between two masses is always attractive.
• Charge can be positive or negative, so electric interaction can be attractive (opposite charges) or repulsive (like charges).

(An alternative valid difference would be that gravity acts on mass while electric acts on charge, or that electric fields can be shielded whereas gravitational fields cannot.)

Key Takeaways

• Both gravitational and electric forces have inverse-square dependence for point sources.
• The crucial qualitative difference: gravity attracts only; electricity can attract or repel.

Common Mistakes

• Writing a “difference” that is not always true (e.g. “electric is stronger” is true in many cases but not a defining property).
• Giving two similarities or two differences.
• Forgetting to specify attraction/repulsion clearly.

Things to Be Careful About

• If you mention equations, the inverse-square dependence must be explicit ($1/r^2$).
• If you mention attraction/repulsion, ensure you link it to charge sign (not to mass).

Background Concept

Two point protons interact via two different inverse-square forces:

1. Electrostatic (Coulomb) force between charges $+e$ and $+e$ separated by $R$:

$$F_e = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{R^2}$$

1. Gravitational force between masses $m_p$ and $m_p$ separated by $R$:

$$F_g = \frac{G m_p^2}{R^2}$$

Because both scale as $1/R^2$, their ratio is independent of $R$.

Understanding the Question

Two protons are separated by distance $R$ in space. You must find the magnitude of:

$$\frac{\text{electric force between protons}}{\text{gravitational force between protons}}$$

So we compare $F_e$ and $F_g$ for the same separation.

Approach

• Write expressions for $F_e$ and $F_g$.
• Form $F_e/F_g$.
• Cancel the common $R^2$ factor.
• Substitute constants ($e$, $m_p$, $G$, and $1/(4\pi\varepsilon_0)$) and calculate in standard form.

Step-by-Step Reasoning

1. Start with the two force laws:

$$F_e = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{R^2}, \qquad F_g = \frac{G m_p^2}{R^2}$$

1. Take the ratio:

$$\frac{F_e}{F_g} = \frac{\left(\frac{1}{4\pi\varepsilon_0}\right)\frac{e^2}{R^2}}{\frac{G m_p^2}{R^2}}$$

1. Cancel $R^2$ (this is why the answer does not depend on separation):

$$\frac{F_e}{F_g} = \frac{\left(\frac{1}{4\pi\varepsilon_0}\right)e^2}{G m_p^2}$$

1. Substitute numerical values:

• $\frac{1}{4\pi\varepsilon_0} = 8.99 \times 10^9\ \text{N m}^2\text{C}^{-2}$
• $e = 1.60 \times 10^{-19}\ \text{C}$
• $G = 6.67 \times 10^{-11}\ \text{N m}^2\text{kg}^{-2}$
• $m_p = 1.67 \times 10^{-27}\ \text{kg}$

$$\frac{F_e}{F_g} = \frac{(8.99\times10^9)(1.60\times10^{-19})^2}{(6.67\times10^{-11})(1.67\times10^{-27})^2}$$

1. Handle powers of ten and the main numbers:

• $(1.60\times10^{-19})^2 = 2.56\times10^{-38}$
• $(1.67\times10^{-27})^2 \approx 2.79\times10^{-54}$
• Numerator: $(8.99\times10^9)(2.56\times10^{-38}) \approx 2.30\times10^{-28}$
• Denominator: $(6.67\times10^{-11})(2.79\times10^{-54}) \approx 1.86\times10^{-64}$

So

$$\frac{F_e}{F_g} \approx \frac{2.30\times10^{-28}}{1.86\times10^{-64}} = 1.24\times10^{36} \approx 1.2\times10^{36}$$

This huge ratio shows the electric repulsion between protons is enormously stronger than their gravitational attraction.

Key Takeaways

• Coulomb’s law and Newton’s law of gravitation have the same $1/R^2$ form for point objects.
• When taking a ratio, the distance cancels, leaving a constant value.
• Careful handling of squares and powers of ten is essential.

Common Mistakes

• Forgetting to square $e$ or $m_p$.
• Not cancelling $R^2$ and incorrectly leaving the ratio dependent on $R$.
• Mixing up $\varepsilon_0$ with $1/(4\pi\varepsilon_0)$.
• Power-of-ten errors when squaring $10^{-19}$ or $10^{-27}$.

Things to Be Careful About

• Use consistent significant figures (typically 2–3 s.f. is appropriate here).
• The ratio is dimensionless; if units appear in your final result, you have not formed the ratio correctly.
• Ensure you use proton mass $m_p$ (not electron mass) and elementary charge $e$.