*Question*

The diagram shows the electric field lines of a positively charged conducting sphere of radius

Points A and B are located on the same field line.

(a) Explain why the electric potential decreases from A to B. [2]

(b) Draw, on the axes, the variation of electric potential *V *with distance *r *from the centre of the sphere. [2]

(c) The concept of potential is also used in the context of gravitational fields. Suggest why scientists developed a common terminology to describe different types of fields. [1]

**Answer/Explanation**

**Ans: **

**a.**

ALTERNATIVE 1

work done on moving a positive test charge in any outward direction is negative ✓

potential difference is proportional to this work «so V decreases from A to B» ✓

ALTERNATIVE 2

potential gradient is directed opposite to the field so inwards ✓

the gradient indicates the direction of increase of V «hence V increases towards the

centre/decreases from A to B» ✓

ALTERNATIVE 3

V = \(\frac{KQ}{R}\)

so as r increases V decreases ✓

V is positive as Q is positive ✓

ALTERNATIVE 4

the work done per unit charge in bringing a positive charge from infinity ✓

to point B is less than point A ✓

** b**

curve decreasing asymptotically for r > R

non – zero constant between 0 and R

** c** to highlight similarities between «different» fields

A planet has radius *R*. At a distance *h *above the surface of the planet the gravitational field strength is *g *and the gravitational potential is *V*.

a.i.

State what is meant by gravitational field strength.

Show that *V *= –*g*(*R *+ *h*).

Draw a graph, on the axes, to show the variation of the gravitational potential *V* of the planet with height *h *above the surface of the planet.

A planet has a radius of 3.1 × 10^{6} m. At a point P a distance 2.4 × 10^{7} m above the surface of the planet the gravitational field strength is 2.2 N kg^{–1}. Calculate the gravitational potential at point P, include an appropriate unit for your answer.

The diagram shows the path of an asteroid as it moves past the planet.

When the asteroid was far away from the planet it had negligible speed. Estimate the speed of the asteroid at point P as defined in (b).

The mass of the asteroid is 6.2 × 10^{12} kg. Calculate the gravitational force experienced by the **planet **when the asteroid is at point P.

**Answer/Explanation**

## Markscheme

a.i.

the **«**gravitational**» **force per unit mass exerted on a point/small/test mass

**[1 mark]**

at height *h *potential is *V* = –\(\frac{{GM}}{{(R + h)}}\)

field is *g *= \(\frac{{GM}}{{{{(R + h)}^2}}}\)

**«**dividing gives answer**»**

*Do not allow an answer that starts with g = –*\(\frac{{\Delta V}}{{\Delta r}}\)* and then cancels the deltas and substitutes **R *+ *h*

*[2 marks]*

correct shape and sign

non-zero negative vertical intercept

**[2 marks]**

*V* = **«**–2.2 × (3.1 × 10^{6} + 2.4 × 10^{7}) =**»** **«**–**»** 6.0 × 10^{7} J kg^{–1}

*Unit is essential*

*Allow eg MJ kg ^{–}*

^{1}*if power of 10 is correct*

*Allow other correct SI units eg m*^{2}*s ^{–}*

^{2}

*, N m kg*

^{–}^{1}

*[1 mark]*

total energy at P = 0 / KE gained = GPE lost

**«**\(\frac{1}{2}\)*mv*^{2} + *mV* = 0 ⇒**»** *v* = \(\sqrt { – 2V} \)

*v* = **«**\(\sqrt {2 \times 6.0 \times {{10}^7}} \) =**»** 1.1 × 10^{4} **«**ms^{–1}**»**

*Award **[3] **for a bald correct answer*

*Ignore negative sign errors in the workings*

*Allow ECF from 6(b)*

*[3 marks]*

*ALTERNATIVE 1*

force on asteroid is **«**6.2 × 10^{12} × 2.2 =**»** 1.4 × 10^{13} **«**N**»**

**«**by Newton’s third law**» **this is also the force on the planet

*ALTERNATIVE 2*

mass of planet = 2.4 x 10^{25} **«**kg**» «**from *V* = –\(\frac{{GM}}{{(R + h)}}\)**»**

force on planet **«**\(\frac{{GMm}}{{{{(R + h)}^2}}}\)**»** = 1.4 × 10^{13} **«**N**»**

*MP2 must be explicit*

*[2 marks]*