Magnetic Fields of Current-Carrying Conductors: A High School Experiment, Study notes of Physics

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EXPERIMENT 1 Magnetic field outside a straight conductor

Objectives  To determine the dependence of magnetic fields on the distance from and current on a conductor.  The following will be investigated: o The magnetic field of a straight conductor as a function of the current. o The magnetic field of a straight conductor as a function of the distance from the conductor. o The magnetic field of two parallel conductors, in which the current is flowing in the same direction, as a function of the distance from one conductor on the line joining the two conductors. o The magnetic field of two parallel conductors, in which the current is flowing in opposite directions, as a function of the distance from one conductor on the line joining the two conductors.

Apparatus Current conductors; Coil, 6 turns; Coil; Clamping device; Iron core; Iron core; Power supply; Teslameter, digital; Hall probe; Current transformer; Digital multimeter; Meter scale; Barrel base; Support rod; Right angle clamp; G-clamp; Connecting cord

Experimental set-up

 The equipment was set up as in diagram.

Experimental procedure

 Current was kept constant at I = 1mA.  Current in opposite directions : Distance, R, was varied from outside one of the conductors (negative side), then in-between the conductors and then the other outside of the other conductor (positive side).  Each time distance was varied; corresponding magnetic field, B, reading was taken from the teslameter.  Graphs of B against R and B against 1 were drawn. 𝑅

conductor to the centre, the B is expected to increase. From the centre to the other conductor, B is expected to decrease. o Graph of B against 𝐼: From equation (3), a linear graph where B increases as I increase is expected.  Current in the same direction: o Graph of B against R: From equation (3), from bigger negative R values to the 1 position of the first conductor, B is expected to increase ( 𝑅 ); hence the graph is to increase. From the other conductor to bigger R values, B is expected to 1 decrease ( 𝑅 ); hence the graph is to decrease. In-between the conductors, the individual currents produce B in opposite directions (from the right hand rule) hence total B is expected to be a sum of the individual Bs, that is, much lower o B values. 1 Graph of B against : From equation (3), from bigger negative R values to the 𝑅 position of the first conductor, B is expected to decrease. From the other conductor to bigger R values, B is expected to increase. In-between, from one conductor to the centre, the B is expected to increase. From the centre to the other conductor, B is expected to decrease.

Results

Current in the same direction

Current in opposite directions

R

/mm

1 / 1 𝑅 𝑚𝑚

B /mT

b (^) 𝑟 2 2

Current in the same direction I /mA

B

/mT 0.6 0. 0.8 0. 1 0. 1.2 0. 1.4 1. 1.6 1.

Analysis Current flowing in opposite directions. B against 1 :

intercept, b = y - mx = 0 .0187;

standard deviation about regression line, 𝑅 (^) 𝑠⃗ − 𝑚 (^2) 𝑠⃗ -50mm <= R <= -5mm: x average, x = ∑ 𝑥 = -0.0586; 𝑁

sr = √^ 𝑦𝑦^ 𝑥𝑥^ = 0.0718; 𝑁− 2 standard deviation of the slope, s (^) = 𝑠⃗𝑟^2 y average, y =

∑ 𝑦 𝑁 = 12.4609;^

m (^) 𝑠⃗ 𝑥𝑥

Sxx = ∑^ 𝑥^2 −

Syy = ∑ 𝑦^2 −

(∑ 𝑥)^2 = 0.0277; 𝑁 (∑ 𝑦)^2 = 0.9426; 𝑁

standard deviation of the intercept, s =𝑠⃗ *√ ∑ 𝑥^2 = 0.0340; 𝑁 ∑ 𝑥 −(∑ 𝑥) Sxy =^ ∑^ 𝑥𝑦^ −^

∑ 𝑥 ∑ 𝑦 = - 0.1579; 𝑁 slope of line, m =

𝑠⃗ 𝑥𝑦^ = -5.7071; 𝑠⃗ 𝑥𝑥

R

/mm

1 / 1 𝑅 𝑚𝑚

B /mT

r

r

Syy = ∑^ 𝑦^2 − (∑ 𝑦)^2 𝑁 = 0.0785;

Current flowing in the same direction. B Sxy =^ ∑^ 𝑥𝑦^ −^

∑ 𝑥 ∑ 𝑦 𝑁 = -0.0038; against

1 : 𝑅 -85mm <= R <= -5mm:

x average, x =

∑ 𝑥 = - 0.0405; 𝑁

slope of line, m =

𝑠⃗ 𝑥𝑦^ = -19.1018; 𝑠⃗ 𝑥𝑥 intercept, b = y - mx = 0.8157; standard deviation about regression line,

y average, y = ∑ 𝑦 𝑁 = 0.5665;^ s^ =^

𝑠⃗ 𝑦𝑦 − 𝑚^2 𝑠⃗^ 𝑥𝑥 𝑁− 2 =^ 0.0451;

Sxx = ∑ 𝑥^2 −

(∑ 𝑥)^2 = 0.0357; 𝑁 (∑ 𝑦)^2 standard deviation of the slope, sm^ =

𝑠⃗ 𝑟

2 𝑠⃗ 𝑥𝑥 Syy = ∑^ 𝑦^2 − (^) 𝑁 = 1.0678; = 3.1989;

Sxy =^ ∑^ 𝑥𝑦^ −^

∑ 𝑥 ∑ 𝑦 = - 0.1895; 𝑁 standard deviation of the intercept,

slope of line, m =

𝑠⃗ 𝑥𝑦^ = -5.3104; s =𝑠⃗ * ∑ 𝑥^2 = 0.0975; 𝑠⃗ 𝑥𝑥 intercept, b = y - mx = 0.3516; standard deviation about regression line,

b (^) 𝑟 (^) 𝑁 ∑ 𝑥 (^2) −(∑ 𝑥)^2

s = √

𝑠⃗ 𝑦𝑦 − 𝑚^2 𝑠⃗^ 𝑥𝑥 = 0.0641; 𝑁− 2

standard deviation of the slope, sm

= 0.3340;

𝑠⃗ 𝑟

2 𝑠⃗ 𝑥𝑥 65mm <= R <= 125mm: x average, x =

∑ 𝑥 = 0.0110;

standard deviation of the intercept, ∑ 𝑥^2 y average, y =

𝑁 ∑ 𝑦 𝑁 = 0.2408; sb =𝑠⃗ (^) 𝑟*√ 𝑁 ∑ 𝑥^2 −(∑ 𝑥)^2

5mm <= R <= 20mm: ∑ 𝑥

Sxx = ∑ 𝑥^2 −

Syy = ∑^ 𝑦^2 −

(∑ 𝑥)^2 𝑁 (∑ 𝑦)^2 𝑁

x average, x = 𝑁 = 0.1042; (^) Sxy = ∑ 𝑥𝑦 − ∑^ 𝑥^ ∑^ 𝑦^ = 0.0044; 𝑁 y average, y =

∑ 𝑦 𝑁 2 = 0.1425;^ slope of line, m =

𝑠⃗ 𝑥𝑦^ = 66.3573; 𝑠⃗ 𝑥𝑥 Sxx =^ ∑^ 𝑥^2 −^

(∑ 𝑥) 𝑁 = 0.0135;^

intercept, b = y - mx = -0.4869; standard deviation about regression line, Syy = ∑ 𝑦^2 − (^) (∑ 𝑦)^2 = 0.0257; 𝑁

s = √

𝑠⃗ 𝑦𝑦 − 𝑚^2 𝑠⃗^ 𝑥𝑥 = 0.0419;

Sxy =^ ∑^ 𝑥𝑦^ −^

∑ 𝑥 ∑ 𝑦 𝑁 = 0.0180;

r (^) 𝑁− 2

slope of line, m =

𝑠⃗ 𝑥𝑦 𝑠⃗ 𝑥𝑥^ = 1.3262;^

standard deviation of the slope, sm = 𝑠⃗ 𝑟^2 𝑠⃗ 𝑥𝑥

intercept, b = y - mx = 0.0044; standard deviation about regression line,

standard deviation of the intercept, √𝑠⃗ 𝑦𝑦−^ 𝑚

(^2) 𝑠⃗ (^) 𝑥𝑥 s =𝑠⃗ * ∑^ 𝑥 2 = 0.0574; sr = = 0.0305; 𝑁− 2 b (^) 𝑟 (^) 𝑁 ∑ 𝑥 (^2) −(∑ 𝑥) 2

standard deviation of the slope, sm

= 0.2620;

𝑠⃗ 𝑟

2 𝑠⃗ 𝑥𝑥

standard deviation of the intercept,

s =𝑠⃗ *

∑ 𝑥^2 = 0.0313; b (^) 𝑟 (^) 𝑁 ∑ 𝑥 (^2) −(∑ 𝑥)^2

25mm <= R <= 45mm:

x average, x =

∑ 𝑥 = 0.0298; 𝑁 y average, y = ∑ 𝑦 𝑁 = 0.246;

Sxx = ∑^ 𝑥^2 −

(∑ 𝑥)^2 𝑁 = 0.0001;

r

Sxx = ∑^ 𝑥^2 − (∑ 𝑥)^2 𝑁 = 0. 7; Syy = ∑^ 𝑦^2 −

(∑ 𝑦)^2 𝑁 = 0.4272; Sxy =^ ∑^ 𝑥𝑦^ −^

∑ 𝑥 ∑ 𝑦 = 0.547; 𝑁 slope of line, m =

𝑠⃗ 𝑥𝑦^ = 0.7814; 𝑠⃗ 𝑥𝑥 intercept, b = y - mx = -0.0112; standard deviation about regression line, s = √

𝑠⃗ 𝑦𝑦 − 𝑚^2 𝑠⃗^ 𝑥𝑥 = 0.0032; 𝑁− 2

Current flowing in opposite directions. B against I :

standard deviation of the slope, sm

= 0.0038;

𝑠⃗ 𝑟

2 𝑠⃗ 𝑥𝑥

R = 10mm: standard deviation of the intercept, sb x average, x =

∑ 𝑥 = 1.1; =𝑠⃗ *√

∑ 𝑥^2 = 0.0045; 𝑁 y average, y =

∑ 𝑦 𝑁

𝑟 (^) 𝑁 ∑ 𝑥 (^2) −(∑ 𝑥) 2

Current in opposite directions Graph of B against R:  -50mm <= R <= -5mm: as expected B increases as the distance to the conductor decreases.  5mm <= R <= 20mm: B decreases from conductor to centre.  25mm <= R <= 40mm: B increases from centre to the conductor.  65mm <= R <= 145mm: B decreases with increasing distance from the conductor.  It can be seen that in-between the conductors the total B is greater than the individual Bs for the conductors. Therefore for currents in opposite direction, the fields are in the same direction at the centre of the conductors and they add up. 1 Graph of B against  

: As expected all these graphs are linear. 𝑅 (^1) -50mm <= R <= -5mm: B decreases with. 𝑅  5mm <= R <= 20mm: B increases from conductor towards centre.  25mm <= R <= 40mm: B decreases from centre to the conductor.  65mm <= R <= 145mm: B increases with

1 . 𝑅

Current in opposite directions Graph of B against R:

 -85mm <= R <= -5mm: as expected B increases as the distance to the conductor decreases.  5mm <= R <= 20mm: B decreases from conductor to centre.  25mm <= R <= 45mm: B increases from centre to the conductor.  65mm <= R <= 125mm: B decreases with increasing distance from the conductor.  It can be seen that in-between the conductors the total B is lower than the individual Bs for the conductors. Therefore for currents in the same direction, the fields are in opposing direction at the centre of the conductors and they cancel out. 1 Graph of B against  

: As expected all these graphs are linear. 𝑅 (^1) -85mm <= R <= -5mm: B decreases with. 𝑅  5mm <= R <= 20mm: B increases from conductor towards centre.  25mm <= R <= 45mm: B decreases from centre to the conductor.  65mm <= R <= 125mm: B increases with 1 . 𝑅

Graph of B against I