Interpretation of gradients and areas under graphs where appropriate.

The idea

Graphs summarise how capacitor quantities change with time. A gradient is the rate of change of the vertical quantity with respect to the horizontal quantity, found by drawing a tangent and dividing the vertical change by the horizontal change. An area under a graph is found by counting squares or by approximating with strips, and it equals the product of the two axis quantities. In capacitor work, a graph of charge Q against time has gradient equal to current I, because I = ΔQ/Δt. A graph of current I against time has area equal to charge Q, because Q = IΔt. A graph of potential difference V against charge Q has gradient equal to 1/C, since V = Q/C, and its area equals the energy stored, since W = ½QV. Always read axis labels and units before choosing a method.

What earns the marks

• Identifies the gradient as the rate of change of the vertical quantity with respect to the horizontal quantity, and states the unit as the vertical unit divided by the horizontal unit.
• Draws a tangent at the required point and reads two well-separated points on the tangent to calculate the gradient, rather than using two data points on a curve.
• Recognises that the area under a graph equals the product of the two axis quantities, and estimates it by counting squares or by summing strips.
• Applies the capacitor relationships: gradient of Q–t gives current I; area under I–t gives charge Q; gradient of V–Q gives 1/C; area under V–Q gives energy stored ½QV.
• Checks that the sign and magnitude of a gradient or area are physically sensible for the process, for example a discharging capacitor has a negative dQ/dt and therefore a negative current.
• Uses consistent SI units throughout, converting millicoulombs to coulombs, microfarads to farads and milliseconds to seconds before calculating.

Common mistakes and how to put them right

• Using two points on a curved graph to find a gradient instead of drawing a tangent at the required point.

  Put it right: Draw a tangent that touches the curve at that point and to read two points on the tangent itself.

• Confusing the roles of gradient and area, for example treating the area under a Q–t graph as current.

  Put it right: Check the units, since the area under Q–t has units C·s, which is not current.

• Forgetting to convert prefixes before calculating, such as using microfarads as farads.

  Put it right: Convert every quantity to SI base or coherent units first, so that 47 μF becomes 47 × 10⁻⁶ F.

Exam tips

• Annotate the graph with the tangent or the counted squares so the examiner can see your method, and state the coordinates you used.
• Quote the gradient or area with its unit, and comment on what the value means physically in the context of the capacitor.
• When estimating an area, show the number of squares and the value of one square, then multiply, so partial credit is clear.