Explain — 5 marks
A student is investigating how the brightness of a light bulb changes with distance. She measures the light intensity at different distances from a lamp. Light intensity is the power of light falling on a surface per unit area. The student finds that as she moves further away from the lamp, the light intensity decreases. She collects the following data: at 1 m the intensity is 100 units, at 2 m the intensity is 25 units, and at 4 m the intensity is 6.25 units.
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(a) Explain what type of proportion exists between light intensity and distance from the lamp.
[2 marks]
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(b) Using the data provided, show that light intensity is inversely proportional to the square of the distance (I ∝ 1/d²).
[2 marks]
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(c) Explain why this inverse square relationship is important when designing lighting systems for large rooms.
[1 mark]
Show mark scheme
- (a) Inverse proportion / intensity decreases as distance increases (1 mark)
- (a) Specifically inverse square proportion / I ∝ 1/d² (1 mark)
- (b) Calculation showing 1² = 1, so I = 100 × 1 = 100 units at d = 1 m (or equivalent check at another distance) (1 mark)
- (b) Calculation showing 2² = 4, so I = 100/4 = 25 units at d = 2 m (or showing 4² = 16, so I = 100/16 = 6.25 at d = 4 m) (1 mark)
- (c) Light intensity falls off rapidly with distance, so multiple light sources are needed to illuminate large areas evenly / lamps must be positioned carefully to avoid dark spots (1 mark)
Define — 5 marks
A lighting engineer is designing an outdoor stadium. The intensity of light from a point source follows an inverse square law relationship with distance. The engineer also needs to consider how the electrical power supplied to the lights is directly proportional to the brightness output required.
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(a) Define what is meant by direct proportion.
[2 marks]
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(b) Define what is meant by inverse proportion.
[2 marks]
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(c) Using the context of the stadium lighting, explain why the light intensity follows an inverse square law rather than direct proportion with distance from the light source.
[1 mark]
Show mark scheme
- (a) When one quantity increases, the other increases by the same factor / in a constant ratio (1 mark)
- (a) The ratio between the two quantities remains constant / y = kx where k is a constant (1 mark)
- (b) When one quantity increases, the other decreases by the same factor / in a constant ratio (1 mark)
- (b) The product of the two quantities remains constant / y = k/x where k is a constant (1 mark)
- (c) Light spreads out over an increasingly large spherical surface area as distance increases / intensity is distributed over 4πr² / the light is not concentrated in a single direction (1 mark)
Calculate — 2 marks
A student is investigating how the resistance of a wire changes with its length. They use a wire made of constantan and measure the resistance at different lengths. The wire has a constant cross-sectional area.
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(a) When the wire is 50 cm long, its resistance is 10 Ω. Calculate the resistance of the same wire when it is 100 cm long.
[1 mark]
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(b) Explain why the resistance changes in the way shown in your answer to part (a).
[1 mark]
Show mark scheme
- (a) Correct answer of 20 Ω (accept 20)
- (a) Working must show recognition that resistance is directly proportional to length, e.g. R ∝ L or 10/50 = R/100
- (b) States that resistance is directly proportional to length
- (b) OR states that when length doubles, resistance doubles
- (b) OR explains that longer wire has more material for charge carriers to collide with
Explain — 3 marks
A student investigates how the brightness of a light bulb changes with distance from a light source. She measures the light intensity at different distances from a lamp and records her results in a table.
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(a) The student finds that when she doubles the distance from the lamp, the light intensity becomes one quarter of its original value. Explain what type of proportion this relationship demonstrates.
[1 mark]
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(b) Explain why light intensity and distance follow this type of relationship rather than a direct proportion.
[2 marks]
Show mark scheme
- (a) This is an inverse proportion / inverse square law (1 mark)
- (b) Light spreads out in all directions from the source (1 mark)
- (b) The intensity is distributed over a larger surface area as distance increases / intensity is inversely proportional to the square of the distance (1 mark)
Describe — 3 marks
A printing company produces brochures. The total cost of printing is directly proportional to the number of brochures ordered. When printing a large order, the company can assign more workers to the job. The time taken to complete the job is inversely proportional to the number of workers assigned.
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(a) Describe the relationship between the total cost and the number of brochures ordered.
[1 mark]
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(b) Describe the relationship between the number of workers assigned to the job and the time taken to complete it.
[2 marks]
Show mark scheme
- (a) As the number of brochures increases, the total cost increases (at a constant rate) OR the ratio of cost to number of brochures is constant
- (b) As the number of workers increases, the time taken decreases
- (b) The product of number of workers and time taken is constant
GCSE Perfect