A-Level Physics October/November 2024 Q2(c)(i): Show that, for any particular value of x, the numerical values of g and F are related b…

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  The gravitational field strength $g$ at a distance $x$ from the Sun (mass $M$) is given by: $g = \frac{GM}{x^2}$

  The radiation flux $F$ at a distance $x$ from the Sun (luminosity $L$) is given by: $F = \frac{L}{4\pi x^2}$

  Rearranging both equations to make $x^2$ the subject: From the equation for $g$: $x^2 = \frac{GM}{g}$ From the equation for $F$: $x^2 = \frac{L}{4\pi F}$

  Since both expressions are equal to $x^2$, we can equate them: $\frac{GM}{g} = \frac{L}{4\pi F}$

  Now, we rearrange to make $g$ the subject: $g \times L = GM \times 4\pi F$ $g = \frac{4\pi G M F}{L}$

  This can be written as: $g = \left( \frac{4\pi GM}{L} \right) F$

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  How the marks are awarded

  • C1 — The first mark is awarded for correctly stating the formula for radiation flux, F = L / 4πx², which relates luminosity to the surface area of a sphere at distance x.
  • M1 — The second mark is for the method of rearranging both the equation for gravitational field strength (g = GM/x²) and the equation for radiation flux to make x² the subject in each case.
  • A1 — The final mark is awarded for equating the two expressions for x² and performing the subsequent algebraic manipulation correctly to arrive at the required final expression.

  Common mistakes

  • Forgetting the '4π' term in the radiation flux equation, incorrectly writing F = L/x². This is a fundamental error in understanding intensity as power per unit area.
  • Making an algebraic error when rearranging the combined equation. A common mistake is cross-multiplying incorrectly, leading to a final expression where variables are in the wrong positions (e.g., L in the numerator).
  • Failing to show the intermediate step of making x² the subject for both equations before equating them, which does not satisfy the 'Show that...' instruction.
  • Confusing the formula for gravitational force (F = GmM/x²) with the formula for gravitational field strength (g = GM/x²).

  Examiner tip: Practice identifying a common variable in two separate physics equations and using algebraic substitution to eliminate it, thereby deriving a new relationship between the remaining quantities.

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